tag:blogger.com,1999:blog-20570391726301029712017-08-18T13:10:40.064-07:00Arxiv BlogElwin Martinhttps://plus.google.com/105415809315866631299noreply@blogger.comBlogger15125tag:blogger.com,1999:blog-2057039172630102971.post-50854931358145736092015-05-18T11:53:00.003-07:002015-05-18T11:53:46.261-07:00Welcome Back/Uncrumpling the Blog<h2>Let me explain...</h2><div>Like many other undergraduates, the authors of this blog suffer from both inexperience and intense workloads. This was never more true than in the past two years, when we both created and proceeded to actively neglect this blog. Having experienced the phenomenon of <i>failure induced by inactivity</i>, or as they say in baseball, "a strikeout looking," I can say that I do not like the taste it leaves in the mouth. </div><div><br /></div><div>Now fortunately, many of us have recently become graduate students. Consider us upgraded, more seasoned, but still active and enthusiastic. We now have among us students of astrophysics, QFT, condensed matter, nonlinear dynamics, biophysics-- to be sure, we have a diverse and capable team. I, at least, will begin writing on this blog, for the exercise if nothing else.<br /></div><div>For future reference, my current work will be related to mesoscopic systems, including granular media, elastic and elasto-plastic sheets, yarns, and just about anything in between that provides interesting physical problems for us to think about.</div><div><br /></div><h2>Now, for some content...</h2><div>Today I want to briefly mention a topic (motivated from a review article, just email me if you want to know more) that is so obvious, so menial, that I am relatively confident that you won't expect it.</div><div><br /></div><div>How does paper crumple?</div><div><br /></div><div>How indeed, you say, it seems trivial. But when you consider the idea of a 2-D system, exposed to a uniform force, it becomes a curiously rich playground: why do ridges and vertices appear? Why does nature seem to want to concentrate all of that uniform stress into lower-dimensional structures, when it usually tries to distribute energy randomly?</div><div><br /></div><div>You've only just begun to scratch the surface. These <i>stress focusing </i>phenomena are present in a wide variety of curious systems, from turbulent flows, to galactic accretion, to dielectric breakdown. When and why does nature decide to put all of its eggs in one basket? From the crumpling perspective, it's easy to tell.</div><div><br /></div><div>Paper is very close to what we call an <i>isometric sheet. </i>This means that paper is much, much easier to bend than it is to stretch. As a consequence, paper tries to eliminate stretching, which requires that it only bend in one direction. Look at a piece of crumpled paper, and you'll realize that this is true. For the most part, shapes on the paper are only curved in one direction, and flat in the other. This is a very hard constraint to satisfy, and what you end up with when you try to confine a large piece of paper in a small area is the familiar network of ridges and vertices that you know as crumpled paper.</div><div><br /></div><div>This post was a bit of a warm-up, and in the future I'll be bringing in more commentary-based posts on articles from various journals. I am breaking the arxiv trend, because I don't prefer to live by rules.</div><div><br /></div><div>Zoop.</div><div><br /></div><div><br /></div>Casey Trimblehttps://plus.google.com/102725875441220327713noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-54673855988018593262014-07-30T13:11:00.002-07:002014-07-30T13:11:57.302-07:00[hep-th] Unification of Inflation and Dark Energy from Spontaneous Breaking of Scale Invariance<a href="http://arxiv.org/abs/1407.6281">Unification of Inflation and Dark Energy</a><br /><div><br /></div><div>This paper proposes a method using two independent non-Riemannian volume forms (integrand of a volume integral) to derive an effective potential for the scalar matter field that is capable of describing both the early universe expansion and dark energy in our universe today. In other words, they are able to derive an effective potential that produces accurate energy densities during the inflationary phase of the universe and our current universe containing dark energy.</div><div><br /></div><div>This paper isn't very difficult, but does require some prerequisites to understand. You should be somewhat familiar with tensor calculus. If you understand what Lorena has been doing in her Numerical Relativity posts then you should be fine with the level of tensors here. Some familiarity with scale transformations would be nice so that you can understand their procedure. An understanding of manifolds would be great, but is not necessary for understanding the primary point of this paper.</div><div><br /></div><div>Repeating the authors' calculations here would be redundant (especially due to the small size of the paper), so I will just mention specific things of interest and some information useful when reading the paper. However, I note that it is impossible for me to teach you all the material required to fully understand this paper. This would require a course in General Relativity.<br /><br />The approach the authors use is to variate an action integral which is pretty standard. The peculiar thing here is that they use two independent non-Riemannian forms instead of one. Following standard procedure for the Lagrangians and measure densities, they obtain the following action </div><div>$$ S = \int{d^{4}x \Phi_{1}(A)[R + L^{(1)}]} + \int{d^{4}x \Phi_{2}(B)[L^{(2)} + \frac{\Phi(H)}{\sqrt{-g}}]} $$</div><div><br />Let's analyze this equation and gain some understanding of what it means. Typically in Classical Mechanics, one is dealing with an action of the form $ S = \int{dt L(\Phi,\partial_{\mu}\Phi)} $, where $L$ is the well known Lagrangian. In Special Relativity the Lagrangian may be rewritten as $ L = \int{d^{3}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $, where $\mathcal{L}$ is the Lagrangian density. This allows us to generalize the action to $ S = \int{d^{4}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $ where we are now integrating over space-time.<br /><br />Now we can make some comparisons between our standard special relativistic action integral and the one given in the paper/above. The terms $R$ and $g$ come from General Relativity. $R$ is the Ricci scalar which gives a measure of the curvature of the geometry in consideration. As a clarification to those who haven't encountered relativity or differential geometry of any sort; $g$ is not Earth's acceleration of gravity, rather it is $\|g_{\mu \nu}\|$ where $g_{\mu \nu}$ is the metric tensor for whatever geometry you would like to use. The measures $\Phi$ are the field strengths of typical gauge fields. This will tell you which field is the primary contributor for specific phenomena.<br /><br />The most difficult part of this paper is where they pull out this gauge transformation that leaves the action invariant and therefore informs us of a symmetry. Gauge transformations can be a pain and aren't very intuitive. Moving along, they variate the action integral and discover two terms that are not invariant under the scale transformation and lead to spontaneous symmetry breaking.<br /><br />In the end, what is obtained is an interesting effective Lagrangian and scalar field potential (see paper). The potential smoothly moves from one flat region at large negative values of the parameter to another flat region at large positive values, as can easily be seen in the figure at the end of the paper.<br /><br />The key to the success of this calculation was the use of two volume forms instead of the typical one used by most physicists. Similar results can be obtained with one volume form, but this leads to nonlinear terms which are obviously ugly. It suffices to say that using two volume forms gets rid of this issue and presents a nice way of calculating the vacuum energy density for both the early universe inflationary phase and our present day dark energy phase.<br /><br />That's about all I have to say about this paper. It's a great read that performs a quick calculation of some pretty cool stuff. I wouldn't get bogged down in the details of the paper if you aren't strong in General Relativity. Rather you should notice how awesome this result is! Hope you enjoyed the post as much as I did.</div>Nicholas Lucashttps://plus.google.com/103830764110712272962noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-54594290987590209342014-07-28T18:47:00.002-07:002014-07-29T14:36:52.399-07:00Numerical Relativity: Black Holes and Gravity - Part 3<h2>Spacetime</h2><div><br /></div><div>The definition of spacetime comes from special relativity. It tells us that different inertial frames have different notions of time. Let's take three Cartesian coordinates (x, y, z). By moving these axes parallel to themselves from some origin, we create new values of x, y, and z. In addition, by moving these coordinates we are creating a notion of time, which is different in each of these inertial frames. This means that inertial frames are spanned by four coordinates (t, x, y, z).<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://www.cv.nrao.edu/course/astr534/images/event.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://www.cv.nrao.edu/course/astr534/images/event.gif" height="170" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Moving reference frames. Graphic from www.cv.nrao.edu.</td></tr></tbody></table><br /><br /></div><h3>Flat Spacetime and the Metric</h3><div><br />Just like in geometry where we use the Pythagorean theorem, $ a^{2} + b^{2} = c^{2} $ relativity uses something somewhat similar called a line element of flat spacetime $$ (\Delta s)^{2} = -(c\Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2} $$ A line element, such as the one above, tells us the change in position vectors as time also changes. Combining these changes gives us the distance between points in spacetime - denoted by $ \Delta s $ . In fact, this is the one used in special relativity because it gives us flat spacetime (which is not true in general relativity). Since we have four coordinates that give us a quantity and a direction, they can be written as vectors and the vectors can be written using index notation as shown below $$ \textbf{a} \cdotp \textbf{b} = (a^{\alpha} \textbf{e}_{\alpha}) \cdotp (b^{\beta} \textbf{e}_{\beta}) = (\textbf{e}_{\alpha} \cdotp \textbf{e}_{\beta}) a^{\alpha} b^{\beta} = \eta_{\alpha \beta} a^{\alpha} b^{\beta} $$ where $ \eta_{\alpha \beta} $ is the metric of flat space time and can be shown as a diagonal matrix<br /><br /><center>\(\eta_{\alpha \beta} =\begin{pmatrix}-c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} =\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} = diag(-1,1,1,1)\)</center><br /><center style="text-align: left;">Here the speed of light is made equal to one in order to make further calculations easier. This metric, called a Minkowsky metric, gives us information of the geometry and structure of spacetime. Although the matrix notation looks messy, it can be very useful for understanding calculations. Instead of writing $ (\Delta s)^{2} = \cdots $ we can just say $$ ds^{2} = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta} $$</center><center style="text-align: left;"><span style="text-align: -webkit-center;">The reason why we changed $ \Delta $ into $ d $ is to show that the values are now changing by infinitesimal amounts. Our $ \alpha 's $ and </span><span style="text-align: -webkit-center;">$ \beta 's $ sum over our four coordinates. I recommend learning about Einstein notation so that the formula above becomes clearer. This notation will prove crucial in the posts of gr-qc publications.</span></center><center style="text-align: left;"><span style="text-align: -webkit-center;"><br /></span></center><center></center></div><h3>Curved Spacetime and the Coordinate Problem</h3><div><br /></div><div><span style="color: #cc0000;">**NOTE: The gravitational constant, G, and the speed of light, c, will be equal to one from now on.** </span></div><div><br /></div><div>Inertial frames are useful because of the special symmetries of flat spacetime. However, in general relativity spacetime is curved and, generally, not symmetric. This means that we cannot take one coordinate system (and I am not talking only about Cartesian or spherical) to simplify the laws; instead, we have to chose a specific system for a specific problem. For example, if we use Schwarzschild coordinates (below)<br />$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dt^2 + \left(1 - \frac{2M}{r} \right)^{-1}dr^2 + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$<br />we will find that it breaks down at r = 2M, 0. If instead we decide to use Eddington-Finkelstein coordinates (below)<br />$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dv^2 + 2dvdr + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$ we will only get infinity at the physical singularity - when at r = 0. This shows us that sometimes, we just need to use a different coordinate system, and there is not one that can be used to solve all problems.<br /><br />Just as we use Maxwell's equations for electromagnetic fields, we use Einstein's equation for gravitational fields. Einstein's equation is a set of ten second-order partial differential equations; they are also nonlinear. This set of equations tell us how a measure of local spacetime curvature is related to a measure of matter energy density. To find a measure of spacetime curvature, we must come up with a thought experiment on the motion of at least two test particles. Imagine two rings of test particles, one is in space and the other is falling towards the Earth. The particles in space feel no force, so they stay in the shape of a ring. However, the ones experiencing the Earth's gravitational force will start forming an ellipse. This occurs because the particles closer to the Earth feel a stronger pull than the ones further away, and the particles on the sides of the falling ring will begin to come closer together since they are pulled towards the center of the Earth. This change of shape in the rings is a measure of spacetime curvature. This is exactly what Newtonian gravity dictates and the <a href="http://sciencejournaljourney.blogspot.com/2014/07/derivation-of-newtonian-deviation.html" target="_blank">derivation</a> is short.<br /><br />Unlike the Newtonian view, Einstein saw that the relative accelerations of these particles are not caused by changes in the gravitational force, but instead, they are caused by the fact that particles simply follow geodesics, or straight lines, in spacetime. And because we are seeing these particles accelerate, their geodesics must be curving relative to each other. This shows us that spacetime is actually being curved. To follow the <a href="http://sciencejournaljourney.blogspot.com/2014/07/derivation-of-equation-of-geodesic.html" target="_blank">derivation</a> of the equation of geodesic deviation in general relativity and understand the implications in-depth, you must understand the use and meaning of Einstein notation, tensors, and Christoffel symbols. You will be able to follow the math without knowing about Christoffel symbols. In the derivations linked below, I will explain as much as I can without going off topic. While the first link is easier to follow, the second link will be mathematically rigorous.<br /><br /><h3>Links from above:</h3><div>1. <a href="http://sciencejournaljourney.blogspot.com/2014/07/derivation-of-newtonian-deviation.html" target="_blank">Derivation of the Newtonian Deviation Equation</a></div><div>2. <a href="http://sciencejournaljourney.blogspot.com/2014/07/derivation-of-equation-of-geodesic.html" target="_blank">Derivation of the Geodesic Deviation Equation</a> </div><br /></div>Lorena Magana Zertuchehttps://plus.google.com/116265683433326275485noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-89549983383474639652014-07-28T18:47:00.001-07:002014-07-28T18:47:04.465-07:00Derivation of the Newtonian Deviation EquationIn this derivation, I will use "A General Relativity Workbook" by Thomas A. Moore as a reference. Below are some basic equations that will help guide us towards the derivation.<br />$$ \textbf{F}_{grav} = m \textbf{g} = - \left( \frac{GmM}{r^2} \right) \textbf{e}_r = -m \boldsymbol{\nabla} \Phi (\textbf{x}) $$<br />where $ \textbf{e}_r $ is the unit vector pointing from the first particle to the other and $ \Phi (\textbf{x}) $ is the gravitational potential from the first particle. Solve for the acceleration in terms of the gravitational potential and you get $ \textbf{a} \equiv \textbf{g} \equiv - \boldsymbol{\nabla} \Phi $ . You can rewrite this using index notation to make it more compact. It should then look like this<br />$$ \frac{d^2 x^i}{dt^2} = - \eta^{ij} \frac{\partial \Phi}{\partial x^j} = - \eta^{ij} [\partial_j \Phi]_\textbf{x} $$<br />Here $ x^i $ is the position of the particle in our frame of reference and $ \eta^{ij} $ represents the spatial components of flat spacetime. This doesn't change the equation but only helps us in balancing out our indices. The equation above tells us how one of our particles falls, but now we need to write a similar equation for the second particle. However, this second particle is at some distance $ \textbf{n} (t) $ away from the first particle; we call this a separation vector. This means that the position of the second particle is at $ x^i(t) + n^i(t) $. With this information we can now write the equation of motion for the second particle, which is<br />$$ \frac{d^2 (x^i + n^i)}{dt^2} = - \eta^{ij} [\partial_j \Phi]_\textbf{x + n} $$<br />Now we must expand the right side in a Taylor series in order to get an estimate of the derivative. This is what the expansion around the reference looks like<br />$$ [\partial_j \Phi]_\textbf{x + n} = [\partial_j \Phi]_\textbf{x} + n^k \left( \frac{\partial}{\partial x^k} [\partial_j \Phi] \right)_\textbf{x} + \cdots $$<br />This works assuming that the quantity of $ \textbf{n} $ is small, so we ignore higher order terms. Now, we are able to substitute this expansion into our equation of motion for the second particle, and then subtract the equation of motion of the first particle from that of the second particle.<br /><br />1) Plug in the Taylor series expansion into the equation of our second particle as follows<br />$$ \frac{d^2 (x^i + n^i)}{dt^2} = - \eta^{ij} [\partial_j \Phi]_\textbf{x} - \eta^{ij} n^k \left( \frac{\partial}{\partial x^k} [\partial_j \Phi] \right)_\textbf{x} $$<br />where<br />$$ \frac{d^2 (x^i + n^i)}{dt^2} = \frac{d^2 x^i}{dt^2} + \frac{d^2 n^i}{dt^2} $$<br />2) Subtract the equation of the first particle from that of the second like this<br />$$ \frac{d^2 x^i}{dt^2} + \frac{d^2 n^i}{dt^2} - \frac{d^2 x^i}{dt^2} = - \eta^{ij} [\partial_j \Phi]_\textbf{x} - \eta^{ij} n^k [\partial_k \partial_j \Phi]_\textbf{x} + \eta^{ij} [\partial_j \Phi]_\textbf{x} $$<br />3) Simplify by canceling out terms to get<br />$$ \frac{d^2 n^i}{dt^2} \approx - \eta^{ij} [\partial_k \partial_j \Phi]_\textbf{x} n^k \equiv - \eta^{ij} \left( \frac{\partial^2 \Phi}{\partial^k \partial^j} \right) n^k $$<br />This is the <b>Newtonian Deviation Equation. </b>By knowing the separation of the two particles at one time and assuming the separation vector remains small, we can determine the deviation of the particles at any given time. <br /><br /><br /><br />Lorena Magana Zertuchehttps://plus.google.com/116265683433326275485noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-76175963681222089502014-07-28T18:47:00.000-07:002014-07-28T18:47:00.533-07:00Derivation of the Equation of Geodesic Deviation<span style="color: #0b5394;">Because this derivation is often given as homework in classes that teach relativity, I will not show step-by-step derivations. Instead, I will only show the steps that books tend to give. If you are confident in your use of tensor notation, you shouldn't have a problem filling in the steps. However, it can get extremely messy when it calls for a change in indices. I will again reference "A General Relativity Workbook" by Thomas A. Moore.</span><br /><br /><div style="text-align: center;">[<a href="http://novicemathandscience.blogspot.com/2012/06/stress-is-rank-2-tensor.html" target="_blank">Here is a "punny" article on tensors.</a>]</div><div style="text-align: center;"><br /></div><div style="text-align: left;">We use tensors in this derivation so that it is valid in arbitrary coordinates. Let's go back to our particles and define their positions by $ x^{\alpha}(\tau) $ and $ \bar{x}^{\alpha}(\tau) \equiv x^{\alpha}(\tau) + n^{\alpha}(\tau) $, where $ \tau $ is our proper time and $ \textbf{n} $ is our infinitesimal separation four-vector. Keep in mind that although this derivation is similar to that in Newtonian mechanics, this derivation uses relativistic concepts. Just as with the Newtonian derivation, we must write expressions that govern the motion of both particles along their geodesics.</div><div style="text-align: left;">$$ 0 = \frac{d^2 x^{\alpha}}{d \tau^2} + \Gamma^{\alpha}_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} , \qquad 0 = \frac{d^2 \bar{x}^{\alpha}}{d \tau^2} + \bar{\Gamma}^{\alpha}_{\mu \nu} \frac{d \bar{x}^{\mu}}{d \tau} \frac{d \bar{x}^{\nu}}{d \tau} $$</div><div style="text-align: left;">Here, we express the particles' coordinate accelerations in terms of the Christoffel symbols instead of having them in terms of the gravitational potential. Christoffel symbols are combinations of first derivatives of the metric that describe effects of parallel transport in manifolds. Just like we did in the Newtonian derivation, we can use Taylor series to expand Christoffel symbol of the second particle around the value of the first particle since the separations are infinitesimal (shown below). </div><div style="text-align: left;">$$ \bar{\Gamma}^{\alpha}_{\mu \nu} (at \: \bar{x}^{\alpha} (\tau)) \approx \Gamma^{\alpha}_{\mu \nu} (at \: x^{\alpha} (\tau)) + n^{\sigma} [ \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu}] (at \: x^{\alpha} (\tau)) $$</div><div style="text-align: left;">Now, we know $ \bar{x}^{\alpha}(\tau) \equiv x^{\alpha}(\tau) + n^{\alpha}(\tau) $ so we will plug this and the expansion above into the equation of motion of the second particle which gives us</div><div style="text-align: left;">$$ 0 = \frac{d^2 (x^{\alpha} + n^{\alpha})}{d \tau^2} + [\Gamma^{\alpha}_{\mu \nu} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu})] \frac{d (x^{\mu} + n^{\mu})}{d \tau} \frac{d (x^{\nu} + n^{\nu})}{d \tau} $$</div><div style="text-align: left;">$$ \qquad \qquad \; \; = \frac{d^2 x^{\alpha}}{d \tau^2} + \frac{d^2 n^{\alpha}}{d \tau^2} + [\Gamma^{\alpha}_{\mu \nu} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu})] \left( \frac{d x^{\mu}}{d \tau} + \frac{d n^{\mu}}{d \tau} \right) \left( \frac{d x^{\nu}}{d \tau} + \frac{d n^{\nu}}{d \tau} \right) $$</div><div style="text-align: left;">We want to simplify this more by multiplying out everything and using our geodesic equations. Doing this will give us</div><div style="text-align: left;">$$ ** \quad 0 = \frac{d^2 n^{\alpha}}{d \tau^2} + 2 \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + n^{\sigma} ( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} u^{\mu}) u^{\mu} u^{\nu} $$</div><div style="text-align: left;">where $ u^{\mu} \equiv \frac{d x^{\mu}}{d \tau} $ is the first particle's four-velocity. Because this is a four-vector we know that its derivative with respect to $ \tau $ is</div><div style="text-align: left;">$$ \left( \frac{d \textbf{n}}{d \tau} \right)^{\alpha} = \frac{d n^{\alpha}}{d \tau} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} n^{\nu} $$</div><div style="text-align: left;">and so we take the second derivative and find it to be</div><div style="text-align: left;">$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \left( \frac{d}{d \tau} \left[ \frac{d \textbf{n}}{d \tau} \right] \right)^{\alpha} = \frac{d}{d \tau} \left( \frac{d n^{\alpha}}{d \tau} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} n^{\nu} \right) + \Gamma^{\alpha}_{\sigma \nu} u^{\sigma} \left( \frac{d n^{\nu}}{d \tau} + \Gamma^{\nu}_{\beta \gamma} u^{\beta} n^{\gamma} \right) $$</div><div style="text-align: left;">$$ = \frac{d^2 n^{\alpha}}{d \tau^2} + \frac{d \Gamma^{\alpha}_{\mu \nu}}{d \tau} u^{\mu} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} \frac{d u^{\mu}}{d \tau} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} u^{\sigma} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} \Gamma^{\nu}_{\beta \gamma} u^{\sigma} u^{\beta} n^{\gamma} $$</div><div style="text-align: left;">Now, this is getting really messy so let us simplify by noticing two key things:</div><div style="text-align: left;">$$ 1. \frac{d \Gamma^{\alpha}_{\mu \nu}}{d \tau} = \frac{d x^{\sigma}}{d \tau} \frac{\partial \Gamma^{\alpha}_{\mu \nu}}{\partial x^{\sigma}} = u^{\sigma} \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} $$</div><div style="text-align: left;">$$ 2. \; Changing \; \sigma \; to \; \mu \; will \; make \; two \; of \; the \; terms \; the \; same. $$</div><div style="text-align: left;">Using both of these in the messy equation we have so far simplifies it to</div><div style="text-align: left;">$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \frac{d^2 n^{\alpha}}{d \tau^2} + \left( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} \right) u^{\sigma} u^{\mu} n^{\nu} + \Gamma^{\alpha}_{\mu \nu} \frac{d u^{\mu}}{d \tau} n^{\nu} + 2 \Gamma^{\alpha}_{\mu \nu} u^{\mu} \frac{d n^{\nu}}{d \tau} + \Gamma^{\alpha}_{\sigma \nu} \Gamma^{\nu}_{\beta \gamma} u^{\sigma} u^{\beta} n^{\gamma} $$</div><div style="text-align: left;">Ok, so now the tricky part comes. We need to do three things: use the geodesic equation of the first particle to eliminate $ \frac{d u^{\mu}}{d \tau} $, use the equation labeled with ** in order to eliminate $ \frac{d^2 n^{\alpha}}{d \tau^2} $, and rename indices to pull out a common factor of $ u^{\sigma} u^{\mu} n^{\nu} $. This will take a while. The trickiest part is definitely changing the indices because there are so many of them. Try to see what you can cancel out and remember the symmetries of the Christoffel symbols. After this, we will get</div><div style="text-align: left;">$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = \left( \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} - \partial_{\nu} \Gamma^{\alpha}_{\mu \sigma} + \Gamma^{\alpha}_{\sigma \gamma} \Gamma^{\gamma}_{\mu \nu} - \Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \sigma} \right) u^{\sigma} u^{\mu} n^{\nu} $$</div><div style="text-align: left;">Since the left side is a tensor and everything outside the parenthesis on the right side is a tensor, then the stuff inside the parenthesis must be a tensor. Not surprisingly, the stuff inside is actually the Riemann tensor, but with the signs flipped. So the Riemann tensor, which is the measure of spacetime curvature, is defined as</div><div style="text-align: left;">$$ R^{\alpha}_{\mu \nu \sigma} \equiv \partial_{\nu} \Gamma^{\alpha}_{\mu \sigma} - \partial_{\sigma} \Gamma^{\alpha}_{\mu \nu} + \Gamma^{\alpha}_{\nu \gamma} \Gamma^{\gamma}_{\mu \sigma} - \Gamma^{\alpha}_{\sigma \gamma} \Gamma^{\gamma}_{\mu \nu} $$</div><div style="text-align: left;">Therefore, we can rewrite the last equation using the Riemann definition and we get</div><div style="text-align: left;">$$ \left( \frac{d^2 \textbf{n}}{d \tau^2} \right)^{\alpha} = - R^{\alpha}_{\mu \nu \sigma} u^{\sigma} u^{\mu} n^{\nu} $$</div><div style="text-align: left;">This is the <b>Geodesic Deviation Equation</b>. This is extremely useful because it allows us to see if spacetime is flat or curved, which we can't do by looking at the metric. So if the any of the components of the Riemann tensor are non-zero, then spacetime is curved.</div>Lorena Magana Zertuchehttps://plus.google.com/116265683433326275485noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-14307350183135274342014-07-26T11:05:00.002-07:002014-07-26T11:11:34.017-07:00Review and application of group theory to molecular systems biology [part 1]Edward A Rietman, Robert L Karp, and Jack A Tuszynski; Theor Biol Med Model. 2011; 8: 21.<br /><a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3149578/">Link</a>.<br /><br />-----<br /><br />My opinion: I love it when physicists stick mathematics where it doesn't belong. This paper is a bit old I know (Elwin wanted recent papers), but I really wanted to read it, and I figured I'd summarize it too while I'm at it for this blog. I'm going to break this up into a few pieces since I want to explain concepts as we go for the reader new to abstract algebra.<br /><br />Any arguments with opinions or analyses in this summary are warmly welcomed.<br /><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">---------------------------------------------------------</span><br />Ratings for part 1 <b>only</b>:<br /><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">General: 9/10</span><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">My familiarity with subject: 7/10</span><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Style: 9/10</span><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Ease (for layman scientist with no algebra background): 7/10?</span><br /><span style="background-color: white; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Length: 7pgs [total paper: 27pgs] (excluding references and images)</span><br /><br />------------------------------------------------<br /><br />According to general biology textbooks, life is characterized by<br /><br /><ul><li>metabolism</li><li>self-maintenance</li><li>duplication involving genetic material</li><li>evolution by natural selection</li></ul><div><br /></div><div>But many complex details of life are overlooked or unaccounted for. The universe may be considered a <a href="http://arxiv.org/ftp/arxiv/papers/0910/0910.2629.pdf">Riemannian resonator</a> and "life can be thought of as some sort of <i>machinery</i> the universe uses to diminish energy gradients" (page 1). The foundations of physics is based in group theory: \(SU(3)\) quarks and \(SU(2)\times U(1)\) for electro-weak interactions.</div><div><br /></div><h3>Group Theory</h3><div>For the reader not well versed in group theory, I suggest following <a href="http://sciencejournaljourney.blogspot.com/2014/07/hep-th-simple-introduction-to-particle.html">Nick's post on particle physics</a>. But I will go over the general concepts anyway.</div><div><br /></div><div>A group is a set of objects that behave in a certain way under some operation (addition, multiplication, etc.). More formally, a <a href="http://en.wikipedia.org/wiki/Group_(mathematics)">group</a> \(G\) is a nonempty set with a binary operation, \(\cdot\), which satisfies:</div><div><ol><li>Closure -- any two elements of \(G\) combined return another element in \(G\).</li><li>Associativity -- although the positions of elements usually matters, the order in which you combine juxtaposed elements does not.</li><li>Identity -- there exists an element that causes no effect when combined with any other element</li><li>Inverses -- For every element, there exists another that may be combined with it to return the identity.</li></ol><div>Or stated even <i>more</i> formally,</div></div><div><ol><li>\(\forall g,g' \in G\ \ g\cdot g' \in G\).</li><li>\(\forall a,b,c \in G\ \ a\cdot(b \cdot c) = (a\cdot b)\cdot c\).</li><li>\(\exists e \in G\ \forall g \in G\ g\cdot e = e\cdot g = g\).</li><li>\(\forall g \in G \ \exists g^{-1} \in G\ gg^{-1} = g^{-1}g = e\).</li></ol><div>However the authors of this paper suspiciously leave out condition 1. Which in my mind is absolutely terrifying when considering a group.</div></div><div><br /></div><div>"Isomorphic" is a concept that basically means that two groups behave the same way, and if the elements of one group were relabeled, they would in fact be identical. This is important because "Any group [...] is isomorphic to a subgroup of matrix groups" (page 4). And a subgroup, just so you know, is exactly what you'd want it to be -- a group inside of another group.</div><div><br /></div><div>Some important groups with associated general properties:</div><div><ul><li><a href="http://en.wikipedia.org/wiki/Orthogonal_group">Orthogonal groups</a> \(O(n)\) -- group of rotations in \(n\)-dimensional Euclidean space including reflections.</li><li>Special orthogonal groups \(SO(n)\) -- groups of rotations in \(n\)-dimensional Euclidean space <i>excluding </i>reflections.</li><li><a href="http://en.wikipedia.org/wiki/Unitary_group">Unitary groups</a> \(U(n)\) -- The inverse of an element is its complex conjugate transpose.</li><li>Special unitary groups \(SU(n)\) -- Inverses are complex conjugate transposes, and the determinant is also \(\pm 1\).</li></ul><div><br /></div></div><h3>Genetic Code</h3><div>Ribosomes take in tRNA (nucleic acids) and output proteins (amino acids). This <a href="http://en.wikipedia.org/wiki/Translation_(biology)">translation</a> of information is done by "codons", which are sections of three nucleic acids. Each codon codes for a specific amino acid. Mathematically, we may regard a codon as the direct product of the set of nucleic acids \(S = \{U, C, A, G\}\) with itself thrice which yields \(4^3 = 64\) possible codons. Now hold on, because this is going to get fun.</div><div><br /></div><div>Since there are not 64 amino acids, many codons code for the <i>same</i> amino acids. We can compile two lists:</div><div><ul><li>\(M_1 = \{ AC, CC, CU, ...\}\)</li><li>\(M_2 = \{ CA, AA, AU, ... \}\)</li></ul><div>The first set \(M_1\) corresponds to doublets whose third nucleic acid <i>doesn't matter</i>. Any nucleic acid following the two in an element from \(M_1\) will not change the resulting amino acid. The second set \(M_2\) corresponds to doublets whose third nucleic acid <i>absolutely</i> matters. Without the third nucleic acid, a sequence from \(M_2\) will not code anything.</div></div><div><br /></div><div>With these sets we may define an operation. Let this operation be switching one letter for another. We have a few possibilities:</div><div><ul><li>\(\alpha: A \leftrightarrow C\) and \(U \leftrightarrow G\).</li><li>\(\beta: A \leftrightarrow U\) and \(C \leftrightarrow G\).</li><li>\(\gamma: A \leftrightarrow G\) and \(U \leftrightarrow C\).</li></ul><div>or written in permutation notation,</div></div><div><div><ul><li>\(\alpha =\begin{pmatrix}A & U & G & C \\ C & G & U & A\end{pmatrix}\)</li><li>\(\beta =\begin{pmatrix}A & U & G & C \\ U & A & C & G\end{pmatrix}\)</li><li>\(\gamma =\begin{pmatrix}A & U & G & C \\ G & C & A & U\end{pmatrix}\)</li></ul><div>With this operation defined, we have a group among our sets. In fact, this group we have defined is isomorphic to something known as the <a href="http://en.wikipedia.org/wiki/Klein_four-group">Klein four group</a>, the group that preserves the symmetries of a rectangle in two dimensions. Two scientists extended this representation to a 4-dimensional hypercube, which looks absolutely crazy:</div></div></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-CpUz6kQLlrE/U9PjWioksWI/AAAAAAAAAso/zn1QjK_eIhU/s1600/hypercube.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-CpUz6kQLlrE/U9PjWioksWI/AAAAAAAAAso/zn1QjK_eIhU/s1600/hypercube.png" height="299" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both;">A subgroup \(N\) of \(G\) is called a <a href="http://en.wikipedia.org/wiki/Normal_subgroup">normal subgroup</a> if any element in \(N\) may be multiplied on the left by some element in \(G\), multiplied on the right by that element's inverse, and still be an element of \(N\), or stated formally, \(N \unlhd G \iff \forall n\in N,\ \forall g \in G\ gng^{-1}\in N\). It's just a fancy subgroup that has absolutely terrible philosophical implications upon closer inspection. The four normal subgroups for the \((K4 \times K4)\) representation shown above are written on page 6.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">With all this work done, we may now see some meaning. We may develop a 64-dimensional hypercube of general genetic code (as stated above) by \(D = \{A, U, G, C\} \otimes \{A, U, G, C\} \otimes \{A, U, G, C\}\). The symmetry operations on this space are the codons. Multiple vertices code for the same amino acid, so our mapping is <a href="http://en.wikipedia.org/wiki/Surjective_function">surjective</a><i>.</i></div><div class="separator" style="clear: both;"><i><br /></i></div><div class="separator" style="clear: both;">If we continue even further and include time evolution, then we have a 65-dimensional differentiable information space manifold \(M[X]\). It is actually postulated that <b>evolution is a geodesic in this information spacetime</b>. Holy shit, right? You should rather be thinking <i>bullshit,</i> but let's continue.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">We may define a metric between species (polynucleotide trajectories) statistically by</div><div class="separator" style="clear: both;">$$ d = \left[ \sum\limits_{\mu} \left(x'^{\mu} - x^{\mu}\right)^2\right]^{\frac{1}{2}} $$</div><div class="separator" style="clear: both;">the regular ol' Euclidean metric. From here we may "see regions of the information-spacetime that have not been explored by evolution" (page 7).</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">We may actually analyze our system in terms of <a href="http://en.wikipedia.org/wiki/Symmetry_breaking">symmetry breaking</a> within a higher dimensional <a href="http://en.wikipedia.org/wiki/Lie_algebra">Lie algebra</a>. From the <a href="http://en.wikipedia.org/wiki/Symplectic_group">symplectic group</a> \(sp(6)\) we may break its symmetry to result in our system.</div><div class="separator" style="clear: both;"></div><ol><li>\(sp(6) \supset \left[sp(4) \otimes su(2)\right]\)</li><li>\(\left[sp(4) \otimes su(2)\right] \supset \left[su(2)\otimes su(2) \otimes su(2)\right]\)</li><li>\(\left[su(2)\otimes su(2) \otimes su(2) \right]\supset\left[ su(2)\otimes u(1) \otimes su(2)\right]\)</li><li>\(\left[su(2)\otimes u(1) \otimes su(2)\right] \supset \left[su(2) \otimes u(1)\right]\)</li><li>\(\left[su(2) \otimes u(1)\right] \supset u(1)\)</li></ol><div>We end on page 7 of 29 (the end of this discussion on codons).</div>Conner Herndonhttps://plus.google.com/107727049489496194560noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-79643345535446848902014-07-24T21:55:00.003-07:002014-07-30T12:22:20.377-07:00Numerical Relativity: Black Holes and Gravity - Part 2<h2>Einstein's Theory of General Relativity</h2><div><br /></div><div>You have probably heard about Einstein's Theory of Relativity. His work on special relativity was published in 1905. This introduced a new framework in physics. He showed that the laws of physics are the same in all non-accelerating reference frame and that the speed of light is constant. This revolutionized different branches of physics and led to his 1915 Theory of General Relativity, or his theory of gravity. Einstein's view was that space and time could be distorted by objects with mass. This spacetime would be able to be stretched and bent. A classic example of this is to imagine a heavy object in the middle of a trampoline. Spacetime is distorted just like the fabric of the trampoline. This would change the way we perceive space and time. Perhaps the most outstanding predictions of this theory are black holes and gravitational waves.<br /><br /><h3>Black Holes (BHs)</h3><div><br /></div>In the media, black holes are erroneously described as giant vacuum cleaners in space. Black holes do not suck in things in the way vacuum cleaners do. Physicist instead think of black holes as regions in space with extreme curvature. Black holes warp space and time in such an extreme manner that nothing - not even light - can escape once it is caught into orbit. Since light can't escape it, we cannot look up at the sky and actually point at a black hole. Instead we use different methods to find them. For example, we look for an object orbiting some dark region of space. We will discuss this further in a future post.<br /><br /><h4>Black Hole Anatomy</h4><div><br /></div>Let's look at the anatomy of a black hole. Although a black hole is not an actual object we can touch, let's think of it as a black sphere; we choose black since we can't see black holes in the sky. First, let's call the surface of the sphere the event horizon. Once anything passes this point, it cannot return outside. In fact, we don't even know what happens inside the event horizon. All that we know is that there is a point inside which we call a singularity. A singularity is when the gravitational field is at infinity. This is where physics breaks down.<br /><br /><h4>How Do Black Holes Form?</h4><br />There are three main ways in which black holes form: implosion, high-energy collision, and binary black hole (BBH) collisions. Implosion, or gravitational collapse, is perhaps the most talked about way of black hole formation. This occurs when a star that is a few times larger than the size of our sun collapse onto itself due to a greater gravitation (or inward) force and smaller internal pressure (outward force). I will mention that for a star to implode, there is a minimum size, which is about 3 solar masses. This Tolman-Oppenheimer-Volkoff limit, however, is not well known because it depends on the equations of state for matter this dense, i.e. the relationship between the volume, pressure, temperature, and internal energy of a star.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://www.quecomoquien.es/files/2008/03/equihidro.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://www.quecomoquien.es/files/2008/03/equihidro.jpg" height="173" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Diagram of gravitational force vs. internal pressure on a star .</td></tr></tbody></table>I won't go into details on the formation by high-energy collisions, since this could only happen in very special conditions. This type of event has never been detected, so it is purely theoretical. If it actually ever happened, the black hole would be so small that it would evaporate extremely fast.<br /><br />Formation by binary black hole collisions is in a way the "easiest" to understand. Unlike implosion where there is an actual birth of a black hole, this type of collisions involve two already formed black holes that collide in order to form a bigger black hole. A very cool thing about this is that the final mass of this black hole is not equal to that of the sum of the two initial black holes. You might be wondering why! The next section gives us the answer.<br /><br /><h3>Gravitational Waves (GWs)</h3><div><br /></div><div>Fantastic! So in the paragraph above we asked why the mass of the final black hole is not the sum of the two initial black holes that collided. This is actually my research area now, so I will try to explain it very thoroughly.<br /><br />Gravitational waves, or gravitational radiation, are a result of acceleration; it is like light only in the sense that both types of radiation carry energy away. A great example of this is a system of two black holes. When two massive bodies orbit around each other, they accelerate. Because our system must conserve energy, gravitational radiation must be given off as our BHs accelerate, and this draws our BHs closer together which causes them to orbit faster. This radiation distorts spacetime, which is why you might have heard of the definition "Gravitational waves are ripples in spacetime." The closer they are, the more strongly radiate.<br /><br /></div><div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://svs.gsfc.nasa.gov/vis/a010000/a010500/a010543/ColWhiteDwarfTV.0538_web.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://svs.gsfc.nasa.gov/vis/a010000/a010500/a010543/ColWhiteDwarfTV.0538_web.png" height="135" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Binary neutron stars orbiting each other and radiating. Photo taken from NASA.</td></tr></tbody></table><br /></div>Below, you can see a picture of this radiation as the black holes get closer to each other and eventually collide. The height is the amplitude of the gravitational waves. the length shows the passing of time. Evidently, gravitational waves are given off as the BHs orbit each other. As they get closer the radiation increases by large factor. The highest point you see in the wave above, shows the time at which the black holes collided to become one. This releases an incredible amount of energy, making a BBH collision the most energetic phenomena in space. After they collide, the final BH will ring like a bell (radiating more) and eventually dying down to a quiescent state. When a black hole is not moving or "eating" anything, it will not radiate. </div><div><br /></div><div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-mTLOkMoIVS4/U9HhOCj3QZI/AAAAAAAAAKI/Bg39Aa7Yizg/s1600/GW.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-mTLOkMoIVS4/U9HhOCj3QZI/AAAAAAAAAKI/Bg39Aa7Yizg/s1600/GW.jpg" height="204" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Gravitational wave from a BBH.</td></tr></tbody></table><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: center;"><tbody><tr><td style="text-align: center;"><br /></td></tr><tr><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table>I will go into more detail on how to detect this waves and what we can learn about them in a future post.</div>Lorena Magana Zertuchehttps://plus.google.com/116265683433326275485noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-74804625408296796392014-07-20T13:27:00.002-07:002014-07-20T18:21:21.272-07:00[hep-ex/th][nucl-ex/th] Motivations behind today's accelerator experiments Part I (General overview)http://arxiv.org/pdf/1306.5009.pdf<br /><br />Level-Undergraduate<br /><br /><a name='more'></a><br /><br /> This 219 page proposal outlines some of the biggest problems in particle/nuclear physics. It is very straight forward (probably because it is meant for a non-scientific audience) description on why we physicist need so much funding to build giant toys...I mean accelerators. I'm going to break this paper up into 4-6 post but I will only discuss the sections which I find are the most important ( you are free to read others that you may be interested in). Everything in this paper is Beyond-Standard Model physics that is on the very edge of our understanding of the universe. It does a very good job of describing why we are interested in each topic and how we are going about answering these questions using the minimal amount of math to explain each case. An important point I would like to make about this paper is the ability to describe complex physical theories to someone not well versed in science. Whenever scientist expand our knowledge in some area (physics,chemistry,etc.) there should be some form of explanation in the format of this paper so that those who are not well acquainted with that particular field have a correct basic understanding of that new idea (instead of the media trying to make headlines by giving out false information). <br /><br /> Some of the key area's of research in this paper include:<br /> -Neutrino oscillations<br /> -The sterile neutrino<br /> -CP violation<br /> -The mass hierarchy of neutrinos<br /> -quark number conservation<br /> -Supersymmetry<br /> -Hadron structure<br /> -Measurement of different EDM's (Electric Dipole Moments)<br />My next post is gonna be about how/why we are going about touching neutrinos so hard. I will also try to find links to papers for each of the individual topics I talk about.<br /><br /><br />Robert Leonardhttps://plus.google.com/106275731658069356372noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-6558356439376798632014-07-20T10:24:00.000-07:002014-07-20T10:24:20.953-07:00[hep-th] A Simple Introduction to Particle Physics <a href="http://arxiv.org/abs/0810.3328">A Simple Introduction to Particle Physics</a><br /><div><br /></div><div>This is a wonderfully written paper introducing the topic of particle physics to undergraduates. It approaches the subject leisurely in Part 1 by reviewing some important classical physics concepts. These include Noether's theorem, gauge transformations, and the Classical Electrodynamics Lagrangian $$ \mathcal{L_{EM}} = -\frac{1}{4} F_{\mu \nu}F^{\mu \nu} - J^{\mu}A_{\mu} $$<br /><br />Following this, Part 2 introduces the algebraic concepts needed to be successful and harness a true understanding of particle physics. Using these foundations, the generators and Lie algebras for any group may be discovered. This portion of the paper is somewhat mathematically rigorous due to the generality of the discussion.<br /><br />The fun really begins in Part 3 of the paper where Quantum Field Theory is finally introduced. The author starts with the typical derivation of the Klein-Gordon equation for spin-0 fields through the relativistic Hamiltonian. Moving forward Robinson et al. discuss spinors and why the irreducible representation of the Lorentz group turns out to be $ SU(2) \otimes SU(2) $. After discussing the Dirac sea of antiparticles and the correct QFT interpretation of antiparticles, we move on to discuss the Dirac Lagrangian for a particle in an electromagnetic field and the coupling of the particle's field (e.g. electron) with the gauge field (photon).<br /><br />Once you reach the section on quantization the paper starts to pick up and get pretty difficult for a reader encountering particle physics for the first time. I suggest only skimming this section or looking up other resources if you have not had experience with this before.<br /><br />Finally, some investigation of the Standard Model is done. The authors look at spontaneous symmetry breaking, the Higgs sector, and the quark sector. You will need to understand what was going on in the quantization section to keep up with their tempo here.<br /><br />This is a fantastic paper that will introduce advanced undergraduates to the beauty and rigor of particle physics without assuming too much early on. The paper becomes difficult in the later portions of Part 3 so is pretty approachable. I hope you all enjoy this paper!<br /><br />P.S. for those of you who highly enjoyed the paper, there is a second paper that goes much deeper into the geometrical approach to gauge theories. </div>Nicholas Lucashttps://plus.google.com/103830764110712272962noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-8997167089972396442014-07-19T23:21:00.000-07:002014-07-20T08:48:51.098-07:00[math.NT] [hep-th] [math-ph] A theory for the zeros of Riemann Zeta and other L-functions. Part - 1 of 3<a href="http://arxiv.org/abs/1407.4358">http://arxiv.org/abs/1407.4358</a><br /><div><br /></div><div>Overview: This paper gives a whopping 97 page overview of the Riemann Zeta function, its impact in physics, and how physics could potentially influence the Riemann-$\zeta$ hypothesis! (If you don't know about this, its importance, or history, you should. It is one of the millennium prize problems. See <a href="http://en.wikipedia.org/wiki/Millennium_Prize_Problems">http://en.wikipedia.org/wiki/Millennium_Prize_Problems</a>) I will certainly not be reviewing the entire work, since some portions delve into number theoretic territory and it is not my intention to cover this. When I first skimmed this paper though, my excitement grew uncontrollably because it gives some wonderful exposition on how this very special function plays a significant role in physics. I thought it would be a great piece to review as my first post since the material can be presented at an approachable level for advanced undergraduates and it has some really deep ideas. Just for a brief taste of what I am talking about, recall that the Riemann Zeta function is defined as </div><div>$$ \zeta (s) := \sum_{n \in \mathbb{N}}\frac{1}{n^s}, \;\;\;\;\; s \in \mathbb{C} $$</div><div>Now because the $\zeta$-function is a function of a complex parameter, we can use techniques of complex analysis to study it and in particular we can use something called <i>analytic continuation</i>. I will explain this concept later when it comes up. But for now, just understand that it almost like a magic wand. Note that $\zeta (-1)$ is divergent (or that's what we <i>think</i>). Well because we can use the magic of analytic continuation, we may <i>extend</i> the $\zeta$-function to another portion of the complex plane where $\zeta (-1)$ gives a finite values. The answer will be $- 1/12$. Look familiar? It should! Because Euler showed that </div><div>$$\sum n = 1 + 2 + 3 + \cdots = -1/12$$</div><div>using some fancy tricks with partial sums and we've all (hopefully) heard this cute story going through our educational careers. But the $\zeta$-function gives us a much deeper understand of why this identity is true (please note that this is not the only way of deriving the identity, ask me if you want to know other ways).</div><div><br /></div><div>But okay...this is nice any cute but why do we physicists care? Because divergences show up ALL the time in physics and we now have a meaningful way of making these sums finite! The general procedure of what I just did is known as $\zeta$-regularization. And guess what, the results we get are physical! (needs citations)</div><div><br /></div><div>Anyway, I just wanted to get everyone reading this interested. Now lets get to the paper and start looking at what it has to offer in the physical realm... which will be in the next post. Coming soon. </div><div><br /></div><div></div>Forrest Kiefferhttps://plus.google.com/107006165054317567255noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-38873198909851561912014-07-19T20:35:00.001-07:002014-07-24T22:12:42.835-07:00Numerical Relativity: Black Holes and Gravity - Part 1<span style="color: blue;">Before getting into the gr-qc publications, I would like to give you some information on this field of study. I recommend knowledge of special relativity, but it is not required. These posts will begin at a low level and become a bit more difficult as we go on. If you are already well acquainted with the concepts and the jargon, you are welcomed to skip them and start at the first gr-qc post.</span><br /><br /><h2>What is Gravity and How Do We Think about It?</h2><div><br /></div><div>Gravity is one of the four fundamental forces. In fact, it is the weakest one! However, an advantage of gravity is that it is a long-range force, which is useful to us when trying to measure interactions. Gravity is the force behind the big bang, black holes, and stars. Gravitational physics is of extreme importance is both large and small scales - from cosmology to quantum physics.</div><div><br />A look into Newtonian gravity will give you the following equation:</div><div>$$ F_{grav} = \frac{Gm_1m_2}{r_{1,2}^2} $$</div><div>which says that the force between two bodies is related to their mass and distance. However, the problem with Newton's view is that the force is instantaneous. This is not allowed, as shown in special relativity where nothing can travel faster than the speed of light. Therefore, we say that Newtonian gravity is an approximation only. </div><div><br /></div><div>We first think of gravity as an accelerating force. For example, in classical physics we are introduced to objects falling from a certain height of being thrown upward at a certain angle. However, when we are exposed to more advanced topics, we begin thinking of gravity as a field with stored energy. However, we can all agree that gravity is geometry. We first study gravity using Euclidean geometry and then go on to learn about non-Euclidean geometry, such as that of the surface of a two dimensional sphere of a radius R. We could go on to talk about different coordinates and invariance, but what we have talked about suffices for now. </div>Lorena Magana Zertuchehttps://plus.google.com/116265683433326275485noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-23671787265425989822014-07-19T16:50:00.002-07:002014-07-20T20:22:12.947-07:00[GR-QC] Negative Mass Bu-...Wait, What?Level: Possibly public, supplemented by undergraduate material<br /><br />So I will begin by acknowledging that I am stepping outside my proposed subfield for this blog a little bit, here, but that this is something that I feel I should address.<br /><br /><a name='more'></a><br /><br /><a href="http://arxiv.org/abs/1407.1457">This preprint</a> [as of yet unpublished!] has been making the rounds in a few internet news sites like <a href="https://news.ycombinator.com/item?id=8043681">Hacker New</a> and <a href="http://science-beta.slashdot.org/story/14/07/17/1334244/cosmologists-show-negative-mass-could-exist-in-our-universe">Slashdot</a>. To begin with, while the arxiv is a wonderful source of information, you <i>do</i> have to take any unpublished results with a proverbial grain of salt, or a truckload, take your pick.<br /><br />For some garbage on the arxiv, you can start with Luboš Motl's examples <a href="http://motls.blogspot.com/2006/05/crackpot-papers-on-arxiv.html">here</a>. For some questionably named content, see Sabine Hossenfelder's Stupid Title List <a href="http://backreaction.blogspot.com/2006/07/stupid-title-list.html">here</a>. While Sabine's list is not necessarily bad science...it gives you some examples of the silliness that you might see. Example of awesome science with a silly title? Escape from the Menace of the Giant Wormholes by Coleman and Lee, Phys. Lett. B 221:242, 1989 :)<br /><br />Abstract [taken verbatim]:<br />We study the possibility of the existence of negative mass bubbles within a de Sitter space-time background with matter content corresponding to a perfect fluid. It is shown that there exist configurations of the perfect fluid, that everywhere satisfy the dominant energy condition, the Einstein equations and the equations of hydrostatic equilibrium, however asymptotically approach the exact solution of Schwarzschid-de Sitter space-time with a negative mass. [Typo on Schwarzschild theirs']<br /><br />Basically they showed that you can form a distribution of stuff with negative mass such that it approaches the classic point distribution solution when you shrink it down, but remains finite otherwise. This is cool, but not actually a particularly challenging calculation nor particularly surprising, if you ask me.<br /><br />One of my favorite lines from the paper: <i>"Thus the negative mass Schwarzschild solution remains an unphysical solution and the question of its meaning is still unanswered."</i><br />Not to be a jerk, but I'm pretty sure the first half of that sentence answers the second half. While tuning parameters into regimes they would not traditionally be allowed into has given some beautiful results in the past (cross section calculations and analytic continuation being my favorite example), having a solution that's mathematically consistent does not imply that it's physically meaningful! There are examples of this everywhere from projectile motion through quantum gravity. The paragraph proceeding that statement effectively lays out an argument showing that a negative mass Schwarzschild solution cannot obey the conditions they desire. As such, they consider a background which is de Sitter instead of Minkowski.<br /><div><br /></div><div>One of the implications that <i>may</i> merit further investigation is that a process which allowed for the production of positive/negative mass pairs in the early universe could allow for a phase of negative mass that might screen gravitational waves. This is something that I will have to think about more. They provided no calculations on the impact of these screening effects; so while there might be a simple mechanism allowing for this, it might be completely washed out by other things.</div><div><br /></div><div>Some background for the uninitiated: </div><div>1. Negative mass in this context is at least naively what you might expect. Their motivation for this study comes from taking the mass parameter, $M$, of the Schwarzschild metric (general relativity's description of a black hole) and extending its domain to the negative numbers. </div><div>$$ ds^2 = - \left( 1 - \frac{2 GM}{r c^2} \right)dt^2 + - \left( 1 - \frac{2 GM}{r c^2} \right)^{-1}dr^2 + r^2 d \Omega $$</div><div><br /></div><div>2. A de Sitter space-time is, roughly, the geometry we believe the universe conforms to on large scales - $\sim 10^9$ parsecs is probably safe if you want a number. It's a lovely space with a constant positive curvature that's simply connected. When you solve Einstein's field equations for a maximally symmetric vacuum with a cosmological constant, $\Lambda$, this is the class of $\Lambda >0$ solutions. A cute fact about de Sitter space is that it was simultaneously and independently discovered by Tullio Levi-Civita of $\epsilon_{ij \ldots n}$ fame. Another cute fact, for the group-theoretically minded, is that the isometry group of de Sitter is $O(1,n)$ for an $n+1$ dimensional space.</div><div><br /></div><div>3. A perfect fluid is completely characterized by its density, $\rho$, and pressure, $p$, in its rest frame. The stress-energy tensor, the object which tells you the densities of the energy and momentum for a system, for such a fluid is given by:</div><div>$$ T^{\mu \nu} = \left( \rho + \frac{p}{c^2}\right) U^\mu U^\nu + p \eta^{\mu \nu} $$</div><div>where $\eta_{mu \nu}$ is the Minkowski metric and and $U$ is the velocity vector field of the fluid. It's worth noting that they're devoid of viscosity and heat conduction.</div><div><br /></div><div>4. The dominant energy condition is a generalization of the weak energy condition (matter density is always non-negative for time-like vectors) that prevents mass-energy from being observed to be flowing faster than light. </div><div>Weak energy condition:</div><div>$$ \rho = T_{ab} X^a X^b \geq 0 $$</div><div>Dominant energy condition: $ - T^a_b Y^b $ must be future-pointing for every timelike or null vector field $Y$.</div><div>NB: Each of these is an ansatz about a system. Einstein's equations are agnostic to a lot of the properties of the matter that you could feed them. While these conditions are things people would consider <i>reasonable</i> they're by no means necessary nor fundamental.</div><div><br /></div><div>5. Hydrostatic equilibrium is basically the requirement that the mechanics of pressures and energy densities are consistent in a system. It's pretty straightforward to arrive at the equations from simply summing forces in terms of the densities and pressures present such that they sum to zero. For example, a simple fluid with a pressure $p$ with density $\rho$, in a local gravitational acceleration $g$ in the $z$ direction gives:</div><div>$$ \frac{\partial p}{\partial z} + \rho g = 0$$</div><div><br /></div><div>6. Finally, the de Sitter-Schwarzschild metric described a black hole sitting in a de Sitter space as described above. It's roughly what the space of a black hole in our universe is expected to look like. The metric is what you might expect, if you're familiar with these things:</div><div>$$ ds^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega $$</div><div>where $f(r) = 1 - \frac{2a}{r} - b r^2$ where $a$ and $b$ are related to the mass of the black hole $M$ and the cosmological constant $\Lambda$.</div><div>These are particularly interesting since they have a maximum size called the <a href="http://link.springer.com/article/10.1023%2FA%3A1026602724948">Nariai spacetime</a> AND they're the simplest objects that have both an event horizon and cosmological horizon. </div><div><br /></div>Elwin Martinhttps://plus.google.com/105415809315866631299noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-31613302088192804382014-07-19T11:42:00.003-07:002014-07-19T11:44:50.978-07:00[q-bio.OT] An Integration of Integrated Information Theory with Fundamental PhysicsSubmitted to the arXiv by Adam B. Barrett on July 3rd, 2014.<br /><a href="http://arxiv.org/abs/1407.4706">http://arxiv.org/abs/1407.4706</a><br /><a href="http://arxiv.org/pdf/1407.4706.pdf">http://arxiv.org/pdf/1407.4706.pdf</a><br /><br />-----<br /><br />My opinion:<br />This paper rings strongly of a physicist entering psychologists' territory in an attempt to steer the course of consciousness research to one of more physical and fundamental origin. Although no equations nor methods of measurement are provided in this paper, the author extends a theory with the intentions of spurring research into the quantitative measurements of fields with respect to consciousness. I was a bit uneasy with the importance placed on consciousness at first, but by the discussion section I was better able to understand the author's intentions.<br /><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">---------------------------------------------------------</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">My familiarity with subject: 5/10</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Style: 8/10</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Ease (for layman): 10/10</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Length: 7pgs (excluding references and images)</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">----------------------------------------------------------</span><br /><br />Any arguments with opinions or analyses in this summary are warmly welcomed. This paper purports what I see to be a possibly controversial issue, and I will try to remain relatively neutral in my synopsis.<br /><br />------------------------------------------------<br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;"><br /></span><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">This paper deals with the subject of consciousness as an incarnation of complex underlying physical and biological mechanisms. Integrated Information Theory (IIT) states that consciousness is compounded information which acts above the sum of its parts. Since non-fundamental information cannot be intrinsic, descriptions of the information in a system depend on an observer's reference point. To surmount this problem, the author provides his solution -- field integrated information hypothesis (FIIH) -- which rests the intrinsic information of a system in the hands of fundamental, physical fields.</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;"><br /></span><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Fields in physics (as opposed to mathematics) are constructs that associate a mathematical object with every point in space and time. All particles have associated fields, and "all forces in nature can be described by field theories which model interactions" (page 4). Since <b>FIIH goes to say that consciousness is a fundamental attribute of matter</b>, the author claims a need to express its manifestation in the behavior of fields/particles.</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;"><br /></span><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">To help quantify consciousness, "the amount of consciousness generated by a patch of field is the amount of integrated information intrinsic to it" (page 4). The quantification must be frame invariant. The author does not delve much further since determining the measurement equation is "beyond the scope of this present paper", but does "speculate" the formula is in terms of "thermodynamic entropy" (page 5).</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;"><br /></span><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">Opponents of IIT criticize the fact that consciousness is produced by integrated information since this continuum approach implies that everyday objects have some consciousness. Since consciousness has multiple definitions depending on the individual, "the key point [...] is that on the theory discussed here, intrinsic integrated information is what underlies subjective experience at the most fundamental level of description" (page 6).</span><br /><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;"><br /></span><span style="background-color: white; color: #444444; font-family: 'Open Sans'; font-size: 15px; line-height: 21px;">The author then continues in a discussion of similar theories of consciousness. One of which he follows with, "this is not however compatible with the laws of physics" (page 7). #rekt. The author goes on to say that although IIT and FIIH do not have yet equations for consciousness, the theories point in the direction of a more quantitative view of the subject.</span>Conner Herndonhttps://plus.google.com/107727049489496194560noreply@blogger.com0tag:blogger.com,1999:blog-2057039172630102971.post-39250936180996520682014-07-17T11:13:00.002-07:002014-07-17T12:30:52.384-07:00[nlin.AO] What about the birds?<b></b><br /><a href="http://arxiv.org/pdf/1407.2414v1.pdf">The Role of Projection in the Control of Bird Flocks</a><br />Pearce, Miller, Rowlands, and Turner<br /><br />Many animals form swarms: cohesive and coherent groups that help individuals gain protection from predators. Some of the most dramatic of these swarms are the murmurations of starlings, shown <a href="https://www.youtube.com/watch?v=eakKfY5aHmY">here.</a> These swarms are usually modeled by assuming that each bird interacts only with a couple of the birds that immediately surround it. These models can describe the shape and structure of large flocks.<br /><br /><a name='more'></a><br /><br />Pearce et. al. focus on a particular aspect of the flocks, the opacity, that they believe are inadequately described by current models. Regardless of the size of the flock, the density of the birds tends to be such that a distant observer can see substantial fractions of both birds and sky through the flocks. The opacity is the percent of the sky visible through the birds. Measured flocks have opacity between 25% and 60%. The opacity also changes immediately before rapid acceleration of the flock, indicating that the opacity allows for long-range information transfer.<br /><br />This level of opacity is evolutionarily favorable. Denser swarms force predators to face target degeneracy, in which it is hard to distinguish an individual target because so many of the birds move together. Sparser swarms allow more of the birds to be able to see an approaching predator, allowing them to aware sooner and evade the predator. Marginal opacity balances both of these benefits.<br /><br />Current models fix the density of the birds by providing interaction models between the birds, creating a fictitious potential between them, or by constraining the volume the birds fly in. However, having a fixed density results in different opacity for different sized flocks. In order to maintain the opacity, these models would have to change their parameters depending on the number of bids in the flock.<br /><br />This paper proposes a hybrid model in which the individual birds respond both to their nearest neighbors and to the density of birds in their line of sight. Birds tend to align themselves with their nearest neighbors: absent other factors, a bird will fly in the direction given by the average velocity of its nearest neighbors. This term leads to collective behavior of the birds. Each bird also responds to its view of the rest of the flock. An individual bird can see dark regions (where its line of sight is intercepted by another bird) and light regions (where it can see through the flock to the sky). Models in which a bird moves towards the lightest or darkest regions it can see result in flock expansion or flock collapse, respectively. Instead, the birds respond to the edges between light and dark regions. Absent other factors, a bird will orient itself towards the average angle of all of the edges it can see. A third term is also included to provide uncorrelated noise to the direction a bird flies.<br /><br />The orientation of a bird depends on three terms: (1) the average orientation of its immediate neighbors, (2) the average direction of the edges of bird and non-bird regions it can see in the rest of the flock, and (3) a stochastic term. The authors numerically explore the parameter space of the relative weights of these three terms.<br /><br />Four numbers are important for each combination of parameters: (1) the opacity of the flock; (2) the maximum distance between any two birds, which measures whether the flock has dispersed; (3) the center of mass velocity of the flock; and (4) the autocorrelation time of the flock, which measures how quickly birds respond to changes on the other edge of the flock. Each of these numbers in averaged over 400,000 time steps for a flock with 100 individuals.<br /><br />If the opacity term is neglected, the flock disperses. The maximum distance between birds is large and the opacity drops to near zero. It is also difficult for the flock to transmit information. The average velocity is small and the autocorrelation time is large.<br /><br />Once the opacity term is included, the flock stabilizes. For most parameter values, the maximum distance between the birds is small and constant. The opacity is consistent with experiments and increases as this term dominates. The average velocity of the flock increases as the correlation between neighboring birds increases relative to the noise term. The autocorrelation time of the flock increases with the opacity term as long as the neighbor interactions are sufficiently strong. <br /><br />The authors also investigated the effects of swarm size. They found that the opacity remains approximately constant as the size of the swarm changes. Larger swarms are slightly more opaque, but the opacity remains less than 75% even for large swarms. This contrasts with other swarming models for which the swarms rapidly become opaque when there are large numbers of birds.<br /><br />Modeling the behavior of large flocks of birds can be described by having each bird respond to a few external visual stimuli. The birds tend to fly in the same direction as their nearest neighbors. Pearce et. al. argue that the birds also respond to the rest of the birds by observing the opacity of the flock. This mechanism results in flocks which have both the target degeneracy to protect individuals from predators and the ability for all of its members to be aware of approaching threats.Jeffrey Heningerhttp://www.blogger.com/profile/15121734816673708102noreply@blogger.com2tag:blogger.com,1999:blog-2057039172630102971.post-16360064002534294452014-07-17T10:15:00.003-07:002014-07-20T09:29:26.049-07:00[q-bio.BM] An Introduction to Biomolecular Simulations and Docking -- Mura and McAnany<a href="http://arxiv.org/abs/1407.3752">http://arxiv.org/abs/1407.3752</a><br /><a href="http://arxiv.org/ftp/arxiv/papers/1407/1407.3752.pdf">http://arxiv.org/ftp/arxiv/papers/1407/1407.3752.pdf</a><br /><br />Any arguments with opinions in this post are welcome. I write this as (hopefully) a guide for those interested in the topic but do not want to go head first into the paper. This paper is a wonderful explanation of molecular dynamics as a tool in computational biophysics, and I highly recommend reading it if you are interested in the field. This is the best paper I've read so far that covers the computational biophysicist's toolset as a whole.<br /><br /><a name='more'></a><br /><br />--------------------------------------------------<br />My familiarity with subject: 9/10<br />Style: 8/10<br />Ease (for layman): 8/10<br />Length: 24pgs (excluding references and images)<br />--------------------------------------------------<br /><br />Sec. 2.2. The paper outlines the necessity, accuracy, and implementation of molecular dynamics simulations as a tool for studying molecular biology. Since "some experimental methods are inherently limited for certain types of questions for any biomolecular system [...], computational approaches such as MD [molecular dynamics] simulation offer an appealing route to exploring [...] full atomic detail, particularly when the desired information is experimentally inaccessible" (page 2). MD simulations probe the region between classical, experimentally verified systems and physical chemistry. This tool for this region is known to the physicist as statistical mechanics.<br /><br />Sec. 3.1. A brief overview of statistical mechanics for the non-physicist: it is "the theoretical framework linking the microscopic (atomic-level) properties of a molecule to its thermodynamic properties" (page 3). In other words, we know quantum mechanics extremely precisely, and we know classical thermodynamics. Statistical mechanics is the link between the two fields. A system of N particles with M states per particle results in \(M^N\) configurations. Since the particles also have kinetic energy, the system Hamiltonian (energy) enjoys a "virtual infinitude of potential configurations". Basically large numbers of particles with interactions, positions, momenta, etc result in a "combinatorial explosion" (page 3). Statistical mechanics takes a probabilistic approach since "any observable/bulk quantity become[s] so strongly spiked that the mean statistical values can be taken as a single, well-defined thermodynamic quantit[y]" (page 4).<br /><br />Sec. 3.2. Now for the interactions on this scale. As for inter-molecular forces, we have electrostatic and van der Waals. The Coulomb force decays as the inverse square of the distance, but (for convenience) the van der Waals force decay is modeled as a Lennard-Jones interaction which decays with the inverse distance to the sixth power. Other forces such as London Dispersion, the hydrophobic effect, and hydrogen bonding can be considered electrostatic in nature. "In summary, electrostatics and vdW forces are what dictate the structure and energetics of biopolymer folding, assembly, and dynamics" (page 6).<br /><br />Sec. 3.4. A degree of freedom (DoF) is a "well-defined parameter that quantifies some property [...] where the parameter is free to vary across a range of values independently of other DoFs" (page 7). For a system with n DoFs, the energy surface is an n-dimensional surface in n+1-dimensional space. A molecule of N atoms in 3-dimensions has 3N DoFs. When simulating a system, we may directly relate the relative populations of our system within configuration space to thermodynamic energy differences (from the Boltzmann distribution). If you picture the population landscape as hills within configuration space, the depth of basins corresponds to enthalpy, and the width to entropy. Although this beautiful portrait of energetics presents itself in theory, adequate sampling of configuration space in reality is difficult.<br /><br />Sec. 3.5. Langevin dynamics is the general approach to this formulation of classical dynamics. The Langevin equation contains a frictional term and a noise term such that in the low friction regime we receive Newton's dynamics, and in the diffusive limit of large friction, we receive Brownian dynamics.<br /><br />It is important to sample all of configuration space. Sometimes deep troughs in the energy landscape go unsampled even though their contribution would proportionately contribute more to the equilibrium ensemble average (by the Boltzmann weight). If we do not sample adequately, we violate the axiom of statistical mechanics "bulk/ensemble properties are calculated from a distribution" (page 9). If we violate this axiom, the tools of statistical mechanics fail to reflect the true properties of the system.<br /><br />Sec. 4.2. How accurate is MD at the scale we're interested? We can base our estimates on the de Broglie wavelength and the Born-Oppenheimer approximation. The thermal de Broglie wavelength is $$\Lambda = \frac{h}{\sqrt{2\pi m k_B T}},$$ where \(h\) is Planck's constant, \(k_B\) Boltzmann's, particle mass \(m\), and temperature \(T\). If \(\Lambda\) is much less than the average inter-particle separation in our system, we're totally cool. Luckily this is true for protein systems at typical temperatures.<br /><br />Electron density moves two orders of magnitude more quickly than the nuclei, so we are able to assume the quantum mechanical qavefunction is separable and can be factorized into nuclear and electronic components. At this point, we absorb the electronic component into the effective interatomic potentials and call it a day.<br /><br />Sec. 4.3. SPEAKING OF WHICH, let's now talk about that wonderful interatomic potential. The potential energy function for the system is called the force-field (FF). This FF contains terms for bonds, angles, dihedrals, impropers, Lennard-Jones interactions, and Coulomb attraction. The constants used in these equations are generated from highly detailed quantum mechanical calculations and experiment.<br /><br />Sec. 4.4. With such a large potential and N atoms, our calculation is of complexity O(\(N^2\)). With fancy computer science we reduce this to O(\(NlogN\)). To move the system forward in time, the FF is calculated at the present time. Then with this information, the negative gradient of the potential gives the force which when applied to each atom yields accelerations. These accelerations are applied and each atom is moved forward in time ~2 femtoseconds. This method is iterated until the desired time is achieved.<br /><br />Sec. 4.5. With methods such as vdW force switching, particle-mesh Ewald, and pair-list distance, we can highly reduce the complexity of the simulations. If you want a more detailed understanding of these methods, read page 13 (it is very well explained). Periodic boundary conditions are applied on all sides on the system so that the atoms see an infinite crystal rather than vacuum. When the system is crafted, a short minimization is undertaken so as to reduce ridiculously high potentials at the start since these potentials could cause the system to crash or explode. You must nurture the system. Love on it. Cradle it. Sometimes I find myself humming gentle lullabies to sooth my systems before their inevitable, violent spasms.<br /><br />Sec. 4.7. The authors detail various methods for analysis such as root mean square deviation and fluctuation (RMSD/RMSF), principal component analysis (PCA), and radial distribution functions. For a detailed account, just read pages 14-16.<br /><br />Sec. 4.7.3. Sometimes particle-mesh Ewald will crash your system if you haven't neutralized it with counter-ions. Good to know.<br /><br />When simulating, there is a choice between computational cost and sampling. For the best results, try to maintain as close to an equilibrium as possible, and make sure your system appears to be hitting all of configuration space.<br /><br />Sec. 4.8. Simulated annealing is a process for better sampling configuration space. The system is heated to ungodly temperatures (so as to provide the necessary thermal energy for surmounting potential mountains), then cooled by a prescribed cooling schedule to reasonable temperatures.<br /><br />Sec. 5. The authors then delved into computational docking. Basically molecules touch sometimes, and it's helpful to know what's going on while that happens. I did not read much further since the paper became much more focused and lost my attention quickly.Conner Herndonhttps://plus.google.com/107727049489496194560noreply@blogger.com0