# Tag Info

14

One important difference between Monte Carlo (MC) and Molecular Dynamics (MD) sampling is that to generate the correct distribution, samples in MC need not follow a physically allowed process, all that is required is that the generation process is ergodic. This can be exploited to accelerate MC schemes. The typical example of this is the Ising model, where ...

13

This is actually a tricky question. First your use of "non equilibrium" is incorrect. Without more information on your MC simulations, especially on the applied biases and simulation process, one cannot state if you are in an out-of-equilibrium (OOE) regime (not state) or not. The way you define it as a state far from the ground state can be a "local" ...

13

I would like to start off by saying this is first and foremost a thermodynamic problem. Secondly, and as a result of thermodynamics, refer to Gibbs Phase Rule which says $$F = C - P + 2$$ Where F = degrees of freedom, C = number of components, and P = number of phases. You seem to be after a pure liquid so $C=1$, and, you are ...

11

I wrote an answer to a similar question in the past, but focused in that question only on the state-of-the-art ultra-high precision calculations on atoms and the three most common isotopologues of $\ce{H_2}$. I will first repeat those here: Atomization energy of the H$_2$ molecule: 35999.582834(11) cm^-1 (present most accurate experiment) 35999.582820(26) cm^...

11

DMC (Diffusion Monte Carlo) Theory. Consider the Schrödinger equation in imaginary time $\tau=it$: $$-\hbar\frac{\partial\psi(x,\tau)}{\partial\tau}=\hat{H}\psi(x,\tau).$$ For a time-independent Hamiltonian $\hat{H}$, the $\tau$-dependence can be solved in a way analogous to the usual time dependence to obtain: $\psi(x,\tau)=\sum_nc_n(0)e^{-E_n\tau/\hbar}\... 10 I could list the models that could be used for microstructural modeling as: Phase-Field: It is constructed based on non-equilibrium thermodynamics and Onsager reciprocal relations to derive a functional for Gibbs free energy and then find the order variables (i.e. phase-field variable to describe the fraction of phases, concentration, temperature, stresses, ... 10 First, some general remarks: The measurements should be made after the system has equilibrated, i.e., a large number of the first iterations should be discarded before the analysis. They should also be averaged over a number of runs, in order to reduce noise. This plot is better appreciated with a log scale in the vertical axis. Later on, it's important to ... 10 I think you are correct that there is an aspect of "take what you can get" to the sizes that are typically used in numerical methods. Even with finite size scaling (FSS), you usually try to go to the largest size that is practical with your computational resources. Case in point: people do finite size scaling with extremely small sizes for exact ... 10 FN-DMC (Fixed-node diffusion Monte Carlo) Theory. See my answer about DMC. The only addition for FN-DMC is that the ground state of an arbitrary Hamiltonian will not be antisymmetrized, and therefore DMC will not converge to the fermionic ground state of interest in electronic systems. To force the system to project out the fermionic ground state, then the ... 9 Most of the problem here appears to be because constraints in the test system default to app.HBonds, which means all bonds to hydrogen atoms are constrained. Something goes very weird when you force a large, unphysical change on the system, and then try to apply HBond constraints (it looks to me like it isn't even constraining HBonds to the unphysical ... 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 @stafusa's answer is great, but there is a specific phenomenon you are encountering here called critical slowing down, which is especially bad for the single-spin-flip Metropolis Algorithm. Near the critical point, the typical cluster size diverges. For the single-spin-flip algorithm, it's really hard to flip these huge clusters, so the autocorrelation ... 8 All depends on what are you working on. When doing a computer simulation, what’s the best way to prepare a starting configuration to avoid biasing your results? If working with molecules in gas phase (isolated systems): you can draw them in your favorite program, save them as 3D model, and then do a simple Molecular Mechanics geometry optimization. you ... 8 The main advantage that MD has is that alot more people have worked on algorithm efficiency. The state of the art codes for MD are really state of the art. Monte Carlo algorithms are fairly primitive in comparison, and many folks just write their own codes. Monte Carlo is better at sampling because it need not follow Newton's equations of motion. If a ... 7 Stochastic Series Expansion (SSE) Monte Carlo Theory: SSE is a finite-temperature, discrete-time technique that works well for quantum spin problems (e.g. Heisenberg model) and other lattice Hamiltonians in any number of dimensions. The method works by expanding the partition function in a Taylor series$$\tag{1} Z = \mathrm{Tr}[ \rho] = \mathrm{Tr}[e^{-\... 7 Your definition is alright, and a citable reference is Landau and Binder's A Guide to Monte Carlo Simulations in Statistical Physics which says (Sec. 2.1.3): The principle of ergodicity states that all possible configurations of the system should be attainable. The usual definition of ergodicity, both in mathematics and physics, is that the ensemble and ... 7 Whatever scattering mechanism you choose must respect detailed balance in equilibrium: on average, the number of particles hitting a patch of the wall at a given angle and velocity must equal the number of particles reflected at the same angle and velocity. If this were not the case, the system would not be in equilibrium. There are numerous ways to do this ... 6 No, nonequilibrium doesn't imply stationarity, but, at the sime time, using "nonequilibrium state" to denote those local energy minima can indeed be confusing, as it might seem to refer to nonequilibrium steady states, which is what your referee is probably getting to. One possible name for these local minima (besides "local minima") could be "non-... 5 Since no one has responded with expertise, I'll attempt a speculative answer here. To my mind, the simplest model of the atoms on the surface of the walls would be an ensemble of independent classical 3D harmonic oscillators at some temperatures$T_{\rm wall}\$. That would be pretty easy to describe from a numerical standpoint, since their velocity could be ...

4

If you want faster solution, you can use simplification. If there are more than some critical amount of spheres (about 100), then grid solution is likely almost as good as true solution. Make a triangle grid, and only check the grid size, adding or removing rows and columns. Add another grid on top, until you have enough spheres to satisfy your requirements. ...

3

Eventually, the question should be more detailed. However, for starters and in general, I would start by giving a cursory read to any wide piece of work that contains this problem, in order to get some perspective. For example in the PhD thesis Cellular automata methods in mathematical physics, where chapter 5 is dedicated to Modeling Polymers with Cellular ...

3

In order to perform a molecular dynamics simulation, you need to equilibrate the system before you can get good statistics. In principle, you can use any starting point, since the simulation should be ergodic; however, this will mean that the equilibration time will be very long. Placing the molecules on a lattice is a simple way to form the starting point, ...

Only top voted, non community-wiki answers of a minimum length are eligible