# Tag Info

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This is an excellent question! Reversibility in MD is useful because: Time-reversibility in a numerical integrator leads to a doubling of the accuracy order (see Propositions 5.2 and Theorem 6.2 here). Reversible maps can be readily Metropolized, for example in a hybrid Monte-Carlo scheme. This gives an easy way to enhance the sampling efficiency and ...

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Another extremely popular resource is Frenkel and Smit's textbook "Understanding Molecular Simulation". It covers all basics on molecular dynamics, Monte Carlo, some common enhanced sampling methods, free energies and it even derives the Ewald summation. This is basically a book that doesn't shy away from messy derivations and it gives you a lot of ...

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A good place to start is the classical Allen-Tildesley book, Computer Simulation of Liquids, which covers the basics of molecular dynamics that hasn't really changed in a long time. The book can be supplemented with literature, such as review articles for whatever it is you want to do. Software manuals are also often quite useful to find out how things are ...

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I would argue the main reason this is important is philosophical, linked to the history of science and determinism (as proposed by Laplace). Newtonian mechanics is mathematically reversable while any observation in a "real world" system is one of irreversibility and increasing entropy, which is why we end up with Loschmidt paradox. From a ...

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Seeing that this question has gathered attention but no replies, I will give it a stab. Note that I am not an expert on DFT or functional calculus, so take this with a grain of salt. As usual, suggestions to the post will be welcome! Using an approach I saw here, we can use a chain rule and obtain the following: $$\frac{\delta F[\rho(\boldsymbol{r})]}{\delta ... 10 Another good book that starts from the very beginning and it's very hands-on is "The art of molecular dynamics" by D. C. Rapaport. It is particularly useful if you want to code up a MD code yourself. This is something I always recommend, even if you want to go on and use one of the big MD packages out there, since you will learn the essentials and ... 10 High frequency in this case is 'particles move all independently, with different directions and speed', and low frequency means 'particles can be represented as gradually changing field of speed and direction of particles in the region'. Something like 'frequency in space', how often to change speed and direction as you observe more and more particles. If ... 9 You are correct that this is due to not including quantum effects. Ref 1 in your figure is the paper cited below. In this paper, they explicitly mention that C_v calculated using the cell-cluster method is in good agreement only for sufficiently high reduced-temperatures. From section IV of this paper: ...the calculation is in acceptable agreement with ... 8 It looks Doi makes some extra simplifications (beyond expanding the exponential) that are valid when the external field is weak. Let's start with what you wrote and make one simplification,$$ \begin{eqnarray} \overline{\delta \phi _a} &= \frac{\langle\delta \phi_a \rangle -\langle \delta \phi _a \beta U_{ext}\rangle}{\langle 1-\beta U_{ext}\rangle} \tag{...

8

Short introduction to ergodicity Ergodicity is when the time-average equals the ensemble-average. A process is ergodic if the time-average "converges in the square mean" to the ensemble average. A sequence $X_t$ converges in the square mean to $X$ if: $$\tag{1} \lim_{t\rightarrow \infty}\langle \left|X_t - X\right|^2 \rangle = 0,$$ where $\... 8 It's very hard to answer this definitively unless you are a bit more specific. However, to speak somewhat generally, it is the case that NVE and NVT ensembles become equivalent in certain thermodynamic limits (namely an infinite number of particles). In practice, however, one chooses the ensemble based on the free energy you are interested in sampling, or ... 8 As with all MD simulations, you have to assume (often wrongly) convergence with finite time. This is fairly easy to do with autocorrelation functions though, because you know that once they become negative, you are already in the random fluctuation (i.e. uncorrelated noise) zone and this is the point where you can stop. This is slightly trickier in some ... 8 It is straightforward to show that in a typical$NPT$setting the Zwanzig equation still only depends on the energy difference and not on the volume (here I define$H$to be the Hamiltonian of each system, respectively and$x$to represent all variables over phase space): $$\frac{Z_{B,NPT}}{Z_{A,NPT}} = \int \int e^{-\beta \Delta H(x)} \frac{e^{-\beta H_A(x)... 7 Thermochemistry (for a single molecule, e.g., ideal gas) depends on the Temperature. At T=0K thermochemistry is kind to us. At T = 0K U = G = H. This is because H = U + PV, which for an ideal gas is H = U + RT, and T = 0K, and also, G = H-TS, and again, T = 0K so that cancels The only direct calculation a QM program is doing, that I am aware of, is ... 7 Energy in an NPT simulation is not conserved, but (once equilibrated), it will fluctuate around an average value, and that average value has meaning. That is the ensemble average for your NPT and is a valuable and useful property. You are also correct that the internal energy is the summation of potential and kinetic contributions. To be thorough, pp. 60 of ... 7 The flying ice cube effect is when the kinetic energy leaks into the translations and rotations. In a constant energy simulation (NVE) this must come at the expense of vibrations, which has the effect of "freezing" the bonds, angles and torsions. Obviously, this is a purely classical effect because even at 0 K a bond will vibrate at the zero-point ... 7 This is a very good and tricky question, which I don't think has a clear and definite answer. I think I should also preface by saying that I can't answer it from the point of view of polymer physics, but instead from my experience in biomolecular simulations. However, this particular problem is basically a version of the sampling exploration-exploitation ... 6 Before working through the equations, I'll try to explain the logic behind what they are doing. It helps to think of think of their process backwards and assume they want an expression like equation 26:$$\mathbf{F}_I(\mathbf{R}^N)=\langle\mathcal{F}_I(\mathbf{r}^n)\rangle_{\mathbf{R}^N} \tag{7}\label{7}$$where the left-hand side is the coarse-grained ... 6 Doi makes this slightly more confusing because just after these equations, he writes: Here for simplicity we have dropped the subscript q. So all of the pieces of your Eqs \eqref{7} and \eqref{8} are actually the Fourier transforms of the pieces of your \eqref{1} and \eqref{2}, so they should all have q subscripts. Its also common to write Fourier ... 6 First, let's look from the most fundamental point of view. For this part of the answer I'll be referencing McQuarrie's Statistical Mechanics [1]. As the Gibbs free energy, G, can be calculated from the molecular partition function, q_{\text {molecule }}, we will start with q_{\text {molecule }}:$$q_{\text {molecule }}=q_{\text {translational }} q_{\... 5 I know it's a boring thing to say, but: It depends on what you want to do. The way I use Langevin thermostats, is to ensure good equipartitioning in my setup, so that I don't have any local hotspots in the system or something like that. If heat transfer is slow in the system, "global" thermostats like Nosé-Hoover and the like will equilibrate ... 5 There are two possible things tripping you up here: Phonons are collective oscillations: they involve the motion of all the atoms together. Therefore it only makes sense to talk about the phonons of the whole system, not any individual atom. The density of states only makes sense as you take$N\to \infty$. For finite$N$, there is a finite/discrete number ... 5 I'll try to summarize the argument mentioned in the comments from Daniel Schroeder's Introduction to Thermal Physics. Your derivation is correct within a certain approximation commonly made when dealing with atomic sized systems. Consider an isolated system of an hydrogen atom in thermal equilibrium with a reservoir (which could in principle be the rest of ... 5 For context to future readers, Doi starts with a model of an$N$unit polymer formed by a random walk along a uniform grid of lattice length$b$. It seems to be implicitly assumed in the described derivation that the lattice used is$b\mathbb{Z}^3$, that is the uniform 3D grid. Note there are other types of lattices for which Eq.$\,(\ref{2})$is not true. ... 5 It definitely can matter. The best thing to do is to consider the experiment you are comparing to, even if that experiment hasn't actually been done yet. In many cases, that means you want to run dynamics in the NPT ensemble (since bench experiments tend to be at fixed pressure and temperature). Here's an example where we know we won't get the same results ... 5 Ensembles are essentially artificial constructs. In the thermodynamic limit (for an infinite system size) and as long as we avoid the neighborhood of phase transitions it is generally believed that there is an equivalence between ensembles. A consequence of the equivalence of ensembles is that the basic thermodynamic properties of a model system may be ... 4 "This works for organic molecules, but what happens when the excited states are closer in energy to the ground state, for example in open-shell molecules or in atoms?" If there's excited states close to the ground state, the approximation you said Gaussian uses, where excited state contributions are neglected, seems not to be such a great idea ... 4 Also worth mentioning Bill Hoover's excellent textbooks, Molecular Dynamics which provide an introduction and a slightly different perspective on molecular dynamics, along with Computational Statistical Mechanics. They are freely downloadable from his website. I also second Tuckerman's "Statistical Mechanics: Theory and Molecular Simulation" as one ... 4 Based on your previous questions, I assume this equation was from Section 1.2.1 of Doi's Polymer Physics. While Doi doesn't explicitly state this anywhere that I can find,$\mathbf{q}$is the label he uses for the momentum/wavevector. I'm more accustomed to seeing this denoted as$\mathbf{k}$, but as Anyon noted in the comments,$\mathbf{q}$may be the more ... 4 The work is a single integral over$|r_1-r_2|$, not a double integral over$r_1$and$r_2$. As you are fixing particle 1, you shouldn't integrate over particle 1. Moreover, the work is$w(r) = \int Fdr$, not$w(r) = \int Frdr$, as you can see from dimensional analysis ($dr$has the dimension of length, too). Therefore, you treat$r_1\$ as constant, integrate ...

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