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I've been taught that when simulating a biomolecule in thermal equilibrium, it's best to use the Langevin thermostat - an algorithm which produces a trajectory, which is a realization of a stochastic process given by the Langevin (stochastic differential) equation. I haven't had a doubt about this algorithm's power to create a sample from the Boltzmann distribution, but what about it's "correctness" regarding time-dependent behaviour of the system? In other words: does the movie we get from a simulation reflect how biomolecules actually behave?

Recently, I've been studying van Kampen's Stochastic Processes in Physics and Chemistry (3rd edition), where on page 56 (chapter III, paragraph 2.) there is a rather philosophical discussion of the stochastic description of microscopic systems. Here's the part that I found most confusing:

Having accepted that the irregular motion of a system may be reformulated as a stochastic process, one is faced with the task of choosing the appropriate process. For a closed, isolated system that is usually done as follows. The microscopic deterministic motion may be represented by a trajectory in the phase space $\Gamma$. Each point $X\in\Gamma$ is, after a time $t$ mapped by the motion into a point $X^t\in\Gamma$, where $X^t = f(X,t)$ is uniquely determined. If one now chooses at some initial time $t=0$ not a single initial state $X$, but a probability density $P(x)$ in $\Gamma$, then $f(x,t)$ is a stochastic process as defined in the preceding section. The initial $P(x)$ is to be chosen so as to reflect the way in which the system was prepared. Any other physical quantity pertaining to the system is a function $Y(X^t)$ of the phase point $X^t$ and has therefore also become a stochastic process $Y(X,t)$.

This is the usual approach to the stochastic description of nonequilibrium behavior and fluctuations. It is the starting point of the derivation of the so-called "generalized Langevin equation" and of the Kubo relations in linear response theory. It was even advocated in the first edition of this book -- but it is wrong. The irregular motion of a Brownian particle can not be related to a probability distribution of some initial state. Rather it is brought about by the surrounding bath molecules and is a vestige of all the variables of the total system that have been ignored in order to obtain an equation for the Brownian particle alone, see IV.a and VIII.3. The proper way of establishing the stochastic description of Brownian motion is therefore the careful elimination of the bath variables from the complete set of microscopic equations of the total system.

(Van Kampen provides a reference: N.G. van Kampen and I Oppenheim, Physica A 138, 231 (1986), to which I haven't got access.)

My question: is it then incorrect to use the Langevin thermostat when we are interested in the actual, time-dependant behaviour of a molecular system? Or, perhaps, the Langevin-equation-based description "isn't that bad" once we immerse the biomolecule in a sufficiently large water basin, thus emulating the complexity of the environment?

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  • $\begingroup$ I have no experience with langevin equations, but I generally accept the notion that "all models are wrong, but some models are useful". Models simplify reality, they don't attempt to replicate it. Models quantify a proposal for what the important effects "might be" under very restrictive circumstances. Either way, the validity of using Langevin versus other models is really more of a biophysics question rather than computational science one. $\endgroup$ – Paul Oct 24 '14 at 23:04
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If $H = H(q, p)$ is the Hamiltonian of your mechanical system, $T$ is the temperature and $\gamma$ is the friction, then Langevin dynamics (LD), \begin{equation*} \left\{ \begin{aligned} \mathrm{d} q &= \nabla_p H(q, p) \, \mathrm{d} t, \\ \mathrm{d} p &= -\nabla_q H(q, p) \, \mathrm{d} t - \gamma \, p \, \mathrm{d} t + \sqrt{2 \gamma k_B T} \, \mathrm{d} B, \end{aligned} \right. \end{equation*} will only coincide with Hamiltonian dynamics (HD) when $\gamma = 0$. This will be the case even in the thermodynamic limit, when the canonical and the microcanonical ensembles are equivalent.

The question, then, is how much does LD diverge from HD. The paper [LNP11] addresses this issue by defining two quantities:

  • The rate at which the system, starting from an initial condition sampled from the canonical distribution at temperature $T_0$, converges to the target temperature $T$.
  • The error in the velocity auto-correlation (VAF) function, defined as the difference between the VAF from LD and that of HD (considered as the reference one).

A quantity known as the efficiency of the thermostat (LD in this case) is then defined as the ratio between the rate of convergence and the error per unit time. One of the conclusions is that LD is not the most efficient thermostatting scheme by this measure or, in other words, there exist thermostats that interfere less with the true dynamics than Langevin dynamics.

[LNP11] Benedict Leimkuhler, Emad Noorizadeh, and Oliver Penrose. Comparing the Efficiencies of Stochastic Isothermal Molecular Dynamics Methods. Journal of Statistical Physics, 143(5):921--942, May 2011. http://link.springer.com/article/10.1007%2Fs10955-011-0210-2

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    $\begingroup$ Thank you for your answer! Could you briefly (maybe in 1-3 sentences) summarize the relevant conclusions of the source you linked to? The Stack Exchange sites tend to discourage short answers that provide link without much explanation, and a summary would make your answer a little more self-contained. $\endgroup$ – Geoff Oxberry Oct 24 '14 at 18:09

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