Langevin dynamics

Real world molecular systems occur in air or solvents, rather than in isolation, in a vacuum. Jostling of solvent or air molecules causes friction, and the occasional high velocity collision will perturb the system. Langevin dynamics attempts to extend molecular dynamics to allow for these effects. Also, Langevin dynamics allows temperature to be controlled as with a thermostat, thus approximating the canonical ensemble.

Langevin dynamics mimics the viscous aspect of a solvent. It does not fully model an implicit solvent; specifically, the model does not account for the electrostatic screening and also not for the hydrophobic effect. For denser solvents, hydrodynamic interactions are not captured via Langevin dynamics.

For a system of $N$ particles with masses $M$, with coordinates $X=X(t)$ that constitute a time-dependent random variable, the resulting Langevin equation is{{cite book | first=Tamar | last=Schlick | authorlink= Tamar Schlick | year=2002 | title=Molecular Modeling and Simulation | publisher=Springer | isbn=0-387-95404-X | page = 480 }}{{cite book | last1=Pastor | first1=R.W. | year = 1994 | chapter = Techniques and Applications of Langevin Dynamics Simulations | editor1-last = Luckhurst | editor1-first = G.R. | editor2-last = Veracini | editor2-first = C.A. | title = The Molecular Dynamics of Liquid Crystals |series=NATO ASI Series|volume=431|pages=85–138 |publisher= Springer |location=Dordrecht|doi=10.1007/978-94-011-1168-3_5 |isbn=978-94-010-4509-4 }}

$$M\backslash ,\backslash ddot\{\backslash mathbf\{X\}\}\; =\; -\; \backslash mathbf\{\backslash nabla\}\; U(\backslash mathbf\{X\})\; -\; \backslash gamma\backslash ,M\backslash ,\backslash dot\{\backslash mathbf\{X\}\}\; +\; \backslash sqrt\{2\backslash ,M\backslash ,\backslash gamma\backslash ,k\_\{\backslash rm\; B\}\; T\}\backslash ,\backslash mathbf\{R\}(t)\backslash ,,$$

where $U(\backslash mathbf\{X\})$ is the particle interaction potential; $\backslash nabla$ is the gradient operator such that $-\backslash mathbf\{\backslash nabla\}\; U(\backslash mathbf\{X\})$ is the force calculated from the particle interaction potentials; the dot is a time derivative such that $\backslash dot\{\backslash mathbf\{X\}\}$ is the velocity and $\backslash ddot\{\backslash mathbf\{X\}\}$ is the acceleration; $\backslash gamma$ is the damping constant (units of reciprocal time), also known as the collision frequency; $T$ is the temperature, $k\_\{\backslash rm\; B\}$ is the Boltzmann constant; and $\backslash mathbf\{R\}(t)$ is a delta-correlated stationary Gaussian process with zero-mean, called Gaussian white noise, satisfying

$$\backslash left\backslash langle\; \backslash mathbf\{R\}(t)\; \backslash right\backslash rangle\; =\; 0$$

$$\backslash left\backslash langle\; \backslash mathbf\{R\}(t)\backslash cdot\backslash mathbf\{R\}(t\text{'})\; \backslash right\backslash rangle\; =\; \backslash delta(t\; -\; t\text{'})$$

Here, $\backslash delta$ is the Dirac delta.

Considering the covariance of standard Brownian motion or Wiener process $W\_t$, we can find that

$$\backslash mathbb\{E\}(W\_tW\_\backslash tau)\; =\; \backslash min(t,\backslash tau)$$

Define the covariance matrix of the derivative as

$$\backslash mathbb\{E\}(\backslash dot\{W\_t\}\backslash dot\{W\_\backslash tau\})\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash tau\}\backslash mathbb\{E\}(W\_tW\_\backslash tau)\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash tau\}\backslash min(t,\backslash tau)\; =\backslash delta(t-\backslash tau)$$

So under the sense of covariance we can say that

$$\{\backslash rm\; d\}W\_t\; =\; \backslash mathbf\{R\}(t)\{\backslash rm\; d\}t$$

Without loss of generality, let the mass $M\; =\; 1$, $\backslash sigma\; =\; \backslash sqrt\{M\backslash gamma\; k\_\{\backslash rm\; B\}T\}$, then the original SDE will become

$$\{\backslash rm\; d\}\backslash dot\{\backslash mathbf\{X\}\}\; =\; -\backslash nabla\; U(\backslash mathbf\{X\})\{\backslash rm\; d\}t-\backslash gamma\; \{\backslash rm\; d\}\{\backslash mathbf\{X\}\}\; +\backslash sqrt\{2\}\backslash sigma\{\backslash rm\; d\}\; \backslash mathbf\{W\}(t)$$

=== Overdamped Langevin dynamics ===

If the main objective is to control temperature, care should be exercised to use a small damping constant $\backslash gamma$. As $\backslash gamma$ grows, it spans from the inertial all the way to the diffusive (Brownian) regime. The Langevin dynamics limit of non-inertia is commonly described as Brownian dynamics. Brownian dynamics can be considered as overdamped Langevin dynamics, i.e. Langevin dynamics where no average acceleration takes place. Under this limit we have $\{\backslash rm\; d\}\backslash dot\{X\}=0$, the original SDE then will becomes

$$\{\backslash rm\; d\}\{\backslash mathbf\{X\}\}\; =-\backslash frac\{1\}\{\backslash gamma\}\backslash nabla\; U(\backslash mathbf\{X\})\{\backslash rm\; d\}t+\backslash frac\{\backslash sqrt\{2\}\backslash sigma\}\{\backslash gamma\}\{\backslash rm\; d\}\; \backslash mathbf\{W\}(t)$$

The translational Langevin equation can be solved using various numerical methods {{Cite journal

|last1=Tanygin

|first1=Bogdan

|last2=Melchionna

|first2=Simone

|title=Comparison of effective and stable Langevin dynamics integrators

|journal=Computer Physics Communications

|volume=299

|pages=109152

|year=2024

|issn=0010-4655

|doi=10.1016/j.cpc.2024.109152 |url=https://www.sciencedirect.com/science/article/pii/S0010465524000754

|language=en

|arxiv=2304.09030

|bibcode=2024CoPhC.29909152T

}}{{Cite journal

|last1=Leimkuhler

|first1=Benedict

|last2=Matthews

|first2=Charles

|title=Robust and efficient configurational molecular sampling via Langevin dynamics

|journal=The Journal of Chemical Physics

|volume=138

|issue=17

|year=2013

|doi=10.1063/1.4802990

|pmid=23656109

|arxiv=1304.3269

|bibcode=2013JChPh.138q4102L

|language=en

}} with differences in the sophistication of analytical solutions, the allowed time-steps, time-reversibility (symplectic methods), in the limit of zero friction, etc.

The Langevin equation can be generalized to rotational dynamics of molecules, Brownian particles, etc. A standard (according to NIST{{Cite web|url=https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=860163|title=Wayback Machine|website=tsapps.nist.gov}}) way to do it is to leverage a quaternion-based description of the stochastic rotational motion.{{Cite journal |last=Omelyan |first=Igor P. |date=1998-01-01 |title=On the numerical integration of motion for rigid polyatomics: The modified quaternion approach |url=https://pubs.aip.org/cip/article/12/1/97/136748/On-the-numerical-integration-of-motion-for-rigid |journal=Computers in Physics |language=en |volume=12 |issue=1 |pages=97–103 |doi=10.1063/1.168642 |issn=0894-1866|arxiv=physics/9901028 |bibcode=1998ComPh..12...97O }}{{Cite journal |last1=Groot |first1=Robert D. |last2=Warren |first2=Patrick B. |date=1997-09-15 |title=Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation |url=https://pubs.aip.org/jcp/article/107/11/4423/478600/Dissipative-particle-dynamics-Bridging-the-gap |journal=The Journal of Chemical Physics |language=en |volume=107 |issue=11 |pages=4423–4435 |doi=10.1063/1.474784 |bibcode=1997JChPh.107.4423G |issn=0021-9606}}

Langevin thermostat{{Cite book |last1=Allen |first1=Michael P. |url=https://books.google.com/books?id=WFExDwAAQBAJ&pg=PP1 |title=Computer Simulation of Liquids |last2=Tildesley |first2=Dominic J. |date=2017-08-15 |publisher=Oxford University Press |isbn=978-0-19-252470-6 |language=en}} is a type of Thermostat algorithm in molecular dynamics, which is used to simulate a canonical ensemble (NVT) under a desired temperature. It integrates the following Langevin equation of motion:

$M\; \backslash ddot\{\backslash mathbf\{X\}\}\; =\; -\backslash nabla\; U(\backslash mathbf\{X\})\; -\; \backslash gamma\; \backslash dot\{\backslash mathbf\{X\}\}\; +\; \backslash sqrt\{2\backslash gamma\; k\_BT\}\; \backslash textbf\{R\}(t)$

$-\backslash nabla\; U(\backslash mathbf\{X\})$ is the deterministic force term; $\backslash gamma$ is the friction coefficient and $\backslash gamma\; \backslash dot\{X\}$ is the friction or damping term; the last term is the random force term ($k\_B$: Boltzmann constant, $T$: temperature). This equation allows the system to couple with an imaginary "heat bath": the kinetic energy of the system dissipates from the friction/damping term, and gain from random force/fluctuation; the strength of coupling is controlled by $\backslash gamma$. This equation can be simulated with SDE solvers such as Euler–Maruyama method, where the random force term is replaced by a Gaussian random number in every integration step (variance $\backslash sigma^2\; =\; 2\backslash gamma\; k\_BT/\; \backslash Delta\; t$, $\backslash Delta\; t$: time step), or Langevin Leapfrog integration, etc. This method is also known as Langevin Integrator.{{Cite web |title=21. Integrators — OpenMM User Guide 8.2 documentation |url=http://docs.openmm.org/latest/userguide/theory/04_integrators.html#integrators-theory |access-date=2025-05-12 |website=docs.openmm.org}}

The overdamped Langevin equation gives

$\{\backslash rm\; d\}\; \backslash mathbf\{x\}\_t\; =\; -\; \backslash frac\{D\}\{k\_BT\}\; \backslash nabla\_\backslash mathbf\{x\}\; U(\backslash mathbf\{x\}\_t)\; \{\backslash rm\; d\}t\; +\; \backslash sqrt\{2D\}\; \{\backslash rm\; d\}\; W\_t$

Here, $D\; =\; k\_B\; T\; /\; \backslash gamma$ is the diffusion coefficient from Einstein relation. As proven with Fokker-Planck equation, under appropriate conditions, the stationary distribution of $\backslash mathbf\; x\_t$ is Boltzmann distribution $p(\backslash mathbf\{x\})\; \backslash propto\; e^\{-U(\backslash mathbf\{x\})/k\_BT\}$.

Since that $\backslash nabla\; \backslash log\; p(\backslash mathbf\{x\})=-\backslash nabla\; U(\backslash mathbf\{x\})/k\_BT$, this equation is equivalent to the following form:

$\{\backslash rm\; d\}\; \backslash mathbf\{x\}\_t\; =\; \backslash epsilon\; \backslash nabla\_\backslash mathbf\{x\}\backslash log\; p(\backslash mathbf\; x\_t)\; \{\backslash rm\; d\}t\; +\; \backslash sqrt\{2\backslash epsilon\}\; \{\backslash rm\; d\}\; W\_t$

And the distribution of $\backslash mathbf\; x\_t\; (t\backslash to\; \backslash infty)$ follows $p(\backslash mathbf\{x\})$. In other words, Langevin dynamics drives particles towards a stationary distribution $p(\backslash mathbf\{x\})$ along a gradient flow, due to the $\backslash nabla\; \backslash log\; p(\backslash mathbf\{x\})$ term, while still allowing for some random fluctuations. This provides a Markov Chain Monte Carlo method that can be used to sample data $\backslash mathbf\; x$ from a target distribution $p(\backslash mathbf\{x\})$, known as Langevin Monte Carlo.

In many applications, we have a desired distribution $p(\backslash mathbf\{x\})$ from which we would like to sample $\backslash mathbf\; x$, but direct sampling might be challenging or inefficient. Langevin Monte Carlo offers another way to sample $\backslash mathbf\; x\; \backslash sim\; p(\backslash mathbf\; x)$ by sampling a Markov chain in accordance with the Langevin dynamics whose stationary state is $p(\backslash mathbf\{x\})$. The Metropolis-adjusted Langevin algorithm (MALA) is an example : Given a current state $\backslash mathbf\; x\_t$, the MALA method proposes a new state $\backslash tilde\{x\}\_\{t+1\}$ using the Langevin dynamics above. The proposal is then accepted or rejected based on the Metropolis-Hastings algorithm. The incorporation of the Langevin dynamics in the choice of $\backslash tilde\{x\}\_\{t+1\}$ provides greater computational efficiency, since the dynamics drive the particles into regions of higher $p(\backslash mathbf\{x\})$ probability and are thus more likely to be accepted. Read more in Metropolis-adjusted Langevin algorithm.

Langevin dynamics is one of the basis of score-based generative models. From (overdamped) Langevin dynamics,

$\{\backslash rm\; d\}\; \backslash mathbf\{x\}\_t\; =\; \backslash epsilon\; \backslash nabla\_\backslash mathbf\{x\}$

\log p(\mathbf x_t) {\rm d}t + \sqrt{2\epsilon} {\rm d} W_t

A generative model aims to generate samples that follow (unknown data distribution) $p(\backslash mathbf\{x\})$. To achieve that, a score-based model learns an approximate score function $\backslash mathbf\{s\}\_\backslash theta(\backslash mathbf\{x\})\; \backslash approx\; \backslash nabla\_\backslash mathbf\{x\}\; \backslash log\; p(\backslash mathbf\{x\})$ (a process called score matching). With access to a score function, samples are generated by the following iteration,{{Cite web |title=Generative Modeling by Estimating Gradients of the Data Distribution {{!}} Yang Song |url=https://yang-song.net/blog/2021/score/ |access-date=2025-05-12 |website=yang-song.net}}{{cite arXiv | eprint=2011.13456 | last1=Song | first1=Yang | last2=Sohl-Dickstein | first2=Jascha | last3=Kingma | first3=Diederik P. | last4=Kumar | first4=Abhishek | last5=Ermon | first5=Stefano | last6=Poole | first6=Ben | title=Score-Based Generative Modeling through Stochastic Differential Equations | date=2020 | class=cs.LG }}

$\backslash mathbf\{x\}\_\{i+1\}\; \backslash gets\; \backslash mathbf\{x\}\_i\; +\; \backslash epsilon\; \backslash nabla\_\backslash mathbf\{x\}\; \backslash log\; p(\backslash mathbf\{x\}\_i)\; +\; \backslash sqrt\{2\backslash epsilon\}\; \backslash mathbf\{z\}\_i,$

\quad i=0,1,\cdots, K

with $\backslash mathbf\{z\}\_i\; \backslash sim\; N(0,1)$. As $\backslash epsilon\; \backslash to\; 0$ and $K\; \backslash to\; \backslash infty$, the generated $\backslash mathbf\{x\}\_K$ converge to the target distribution $p(\backslash mathbf\; x)$. Score-based models use $\backslash mathbf\{s\}\_\backslash theta(\backslash mathbf\{x\})\; \backslash approx\; \backslash nabla\_\backslash mathbf\{x\}\; \backslash log\; p(\backslash mathbf\{x\})$ as an approximation.

As a stochastic differential equation(SDE), Langevin dynamics equation, has its corresponding partial differential equation(PDE), Klein-Kramers equation, a special Fokker–Planck equation that governs the probability distribution of the particles in the phase space. The original Langevin dynamics equation can be reformulated as the following first order SDEs:

$$\{\backslash rm\; d\}\backslash mathbf\{X\}\; =\; \backslash mathbf\{P\}\{\backslash rm\; d\}\; t$$

$$\{\backslash rm\; d\}\; \backslash mathbf\{P\}\; =\; -\backslash gamma\backslash mathbf\{P\}\{\backslash rm\; d\}t-\backslash nabla\; U(\backslash mathbf\{X\})\{\backslash rm\; d\}\; t+\backslash sqrt\{2\}\backslash sigma\; \{\backslash rm\; d\}\backslash mathbf\{W\}(t)$$

Now consider the following cases and their law of $(\backslash mathbf\{X\},\backslash mathbf\{P\})$:

1.$\backslash mathbf\{\{\backslash rm\; d\}\{X\}\}\; =\; \backslash mathbf\{P\}\{\backslash rm\; d\}\; t$

, \mathbf{{\rm d}{P}} = -\gamma\mathbf{P}{\rm d} t-\nabla U(\mathbf{X}){\rm d} t+\sqrt{2}\sigma {\rm d}\mathbf{W}(t) with $(\backslash mathbf\{X\}\_0,\backslash mathbf\{P\}\_0)\backslash sim\backslash rho\_0$

2. $\backslash frac\{\backslash partial\backslash rho\}\{\backslash partial\; t\}\; =\; -\backslash mathbf\{P\}\backslash nabla\_\{\backslash mathbf\{X\}\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{P\}\}(\backslash gamma\backslash mathbf\{P\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{X\}\}U(\backslash mathbf\{X\})\backslash rho)+\backslash nabla\_\{\backslash mathbf\{P\}\}^2(\backslash sigma\_\{T\}^2\backslash rho)$ with $\backslash rho(t=0,\backslash mathbf\{X\},\backslash mathbf\{P\})=\backslash rho\_0$

Consider a general function of momentum and position

$$\backslash Psi\_t\; =\; \backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})$$

The expectation value of the function will be

$$\backslash mathbb\{E\}[\backslash Psi\_t]\; =\; \backslash int\; \backslash rho(t,\backslash mathbf\{X\},\backslash mathbf\{P\})\backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})\{\backslash rm\; d\}\; \backslash mathbf\{P\}\{\backslash rm\; d\}\; \backslash mathbf\{X\}$$

Taking derivative with respect to time $t$, and applying Itô's formula, we have

$$\backslash mathbb\{E\}[\backslash frac\{\backslash rm\; d\}\{\{\backslash rm\; d\}t\}\backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})]$$

=\mathbb{E}[\nabla_{\mathbf{X}} \Psi\frac{{\rm d}\mathbf{X}}{{\rm d}t} + \nabla_{\mathbf{P}}\Psi\frac{{\rm d}\mathbf{P}}{{\rm d}t} + \sigma_T^2\nabla_{\mathbf{P}}^2\Psi\frac{1}{{\rm d} t}({\rm d} \mathbf{W}(t))^2]

which can be simplified to

$$\backslash int\; (\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash rho)\backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})\{\backslash rm\; d\}\; \backslash mathbf\{X\}\{\backslash rm\; d\}\backslash mathbf\{P\}\; =\backslash mathbb\{E\}[(\backslash nabla\_\{\backslash mathbf\{X\}\}\; \backslash Psi)\backslash mathbf\{P\}\; +\; \backslash nabla\_\{\backslash mathbf\{P\}\}\backslash Psi(-\backslash gamma\backslash mathbf\{P\}-\backslash nabla\_\{\backslash mathbf\{X\}\}U(\backslash mathbf\{X\}))\; +\; \backslash sigma\_T^2\backslash nabla\_\{\backslash mathbf\{P\}\}^2\backslash Psi]$$

Integration by parts on right hand side, due to vanishing density for infinite momentum or velocity we have

$$(\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash rho)\backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})\{\backslash rm\; d\}\; \backslash mathbf\{X\}\{\backslash rm\; d\}\backslash mathbf\{P\}\; =\; \backslash int(-\backslash mathbf\{P\}\backslash nabla\_\{\backslash mathbf\{X\}\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{P\}\}(\backslash gamma\backslash mathbf\{P\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{X\}\}U(\backslash mathbf\{X\})\backslash rho)+\backslash nabla\_\{\backslash mathbf\{P\}\}^2(\backslash sigma\_\{T\}^2\backslash rho)\; )\backslash Psi(\backslash mathbf\{X\},\backslash mathbf\{P\})\{\backslash rm\; d\}\backslash mathbf\{X\}\{\backslash rm\; d\}\backslash mathbf\{P\}$$

This equation holds for arbitrary $\Psi$, so we require the density to satisfy

$$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; =\; -\backslash mathbf\{P\}\backslash nabla\_\{\backslash mathbf\{X\}\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{P\}\}(\backslash gamma\backslash mathbf\{P\}\backslash rho+\backslash nabla\_\{\backslash mathbf\{X\}\}U(\backslash mathbf\{X\})\backslash rho)+\backslash nabla\_\{\backslash mathbf\{P\}\}^2(\backslash sigma\_\{T\}^2\backslash rho)$$

This equation is called the Klein-Kramers equation, a special version of Fokker Planck equation. It's a partial differential equation that describes the evolution of probability density of the system in the phase space.

For the overdamped limit, we have $\{\backslash rm\; d\}\backslash mathbf\{P\}\; =\; 0$, so the evolution of system can be reduced to the position subspace. Following similar logic we can prove that the SDE for position,

$$\{\backslash rm\; d\}\; \backslash mathbf\{X\}\; =\; -\backslash frac\{1\}\{\backslash gamma\}\backslash nabla\; U(\backslash mathbf\{X\})\{\backslash rm\; d\}\; t\; +\backslash sqrt\{2\}\backslash frac\{\backslash sigma\}\{\backslash gamma\}\backslash mathbf\{R\}(t)\{\backslash rm\; d\}\; t$$

corresponds to the Fokker Planck equation for probability density

$$\backslash frac\{\backslash partial\; \backslash rho(t,\backslash mathbf\{X\})\}\{\backslash partial\; t\}\; =\; \backslash nabla\_\{\backslash mathbf\{X\}\}(\backslash frac\{1\}\{\backslash gamma\}\backslash nabla\_\{\backslash mathbf\{X\}\}U(\backslash mathbf\{X\})\backslash rho(t,\backslash mathbf\{X\}))+\backslash Delta\_\backslash mathbf\{X\}(\backslash frac\{\backslash sigma^2\}\{\backslash gamma^2\}\backslash rho(t,\backslash mathbf\{X\}))$$

Consider Langevin dynamics of a free particle (i.e. $U(\backslash mathbf\{X\})=0$), then the equation for momentum will become

$$\{\backslash rm\; d\}\backslash mathbf\{P\}\; =\; -\backslash frac\{1\}\{\backslash gamma\}\; \backslash mathbf\{P\}\{\backslash rm\; d\}t\; +\backslash frac\{\backslash sqrt\{2\}\backslash sigma\}\{\backslash gamma\}\; \{\backslash rm\; d\}\backslash mathbf\{W\}\_t$$

the analytical solution to this SDE is

$$\backslash mathbf\{P\}\; =\; \backslash mathbf\{P\}\_0e^\{-\; t/\backslash gamma\}+\backslash frac\{\backslash sqrt\{2\}\backslash sigma\}\{\backslash gamma\}\backslash int\_0^t\; \{\backslash rm\; e\}^\{-(t-t\text{'})/\backslash gamma\}\{\backslash rm\; d\}\backslash mathbf\{W\}\_t\text{'}$$

thus the average value of second moment of momentum will becomes (here we apply the Itô isometry)

$$\backslash mathbb\{E\}(\backslash mathbf\{P\}^2)\; =\; \backslash mathbf\{P\}^2\_0\{\backslash rm\; e\}^\{-2\; t/\backslash gamma\}+\; \backslash frac\{\backslash sigma^2\}\{\backslash gamma\}(1-\{\backslash rm\; e\}^\{-2t/\backslash gamma\})\backslash overset\{t\backslash to\backslash infty\}\{\backslash to\}\backslash frac\{\backslash sigma^2\}\{\backslash gamma\}$$

That is, the limiting behavior when time approaches positive infinity, the momentum fluctuation of this system is related to the energy dissipation (friction term parameter $\backslash gamma$) of this system. Combining this result with Equipartition theorem, which relates the average value of kinetic energy of particles with temperature

$$\backslash langle\; v^2\backslash rangle\; =\; k\_\{\backslash rm\; B\}T$$

we can determines the value of the variance $\backslash sigma$ in applications like Langevin thermostat.

$$\backslash sigma^2/\backslash gamma\; =\; k\_B\; T\; \backslash to\; \backslash sigma\; =\; \backslash sqrt\{k\_\{\backslash rm\; B\}\; T\backslash gamma\}$$

This is consistent with the original definition assuming $M=1$.

Path Integral Formulation comes from Quantum Mechanics. But for a Langevin SDE we can also induce a corresponding path integral. Considering the following Overdamped Langevin equation under, where without loss of generality we take $\backslash gamma\; =\; \backslash sigma\; =\; 1$,

$$\{\backslash rm\; d\}\{X\}\; =\; -\backslash nabla\; U(\{X\})\{\backslash rm\; d\}t\; +\backslash sqrt\{2\}\{\backslash rm\; d\}W\_t$$

Discretize and define $t\_n\; =\; n\backslash Delta\; t$, we get

$$\{X\}\_\{n+1\}\; -\{X\}\_\{n\}\; +\; \backslash nabla\; U(\{X\})\backslash Delta\; t=\; \backslash sqrt\{2\}(W\_\{t\_n\}-W\_\{t\_\{n-1\}\})\backslash sim\; \backslash mathcal\{N\}(0,2\backslash sqrt\{\backslash Delta\; t\})$$

Therefore the propagation probability will be

$$P(\{X\}\_\{n+1\}|\{X\}\_\{n\})\; =\; \backslash int\; \{\backslash rm\; d\}\backslash xi\; \backslash frac\{1\}\{2\backslash sqrt\{\backslash pi\backslash Delta\; t\}\}\{\backslash rm\; e\}^\{-\backslash frac\{\backslash xi^2\}\{4\backslash Delta\; t\}\}\; \backslash delta(\{X\}\_\{n+1\}\; -\{X\}\_\{n\}\; +\; \backslash nabla\; U(\{X\})\backslash Delta\; t-\backslash xi)$$

Applying Fourier Transform of delta function, and we will get

$$P\; =\; \backslash int\; \backslash frac\{\{\backslash rm\; d\}k\}\{2\backslash pi\}\{\backslash rm\; e\}^\{\{\backslash rm\; i\}k(\{X\}\_\{n+1\}-\{X\}\_n\; +\; \backslash nabla\; U(\{X\})\backslash Delta\; t)\}\backslash int\; \{\backslash rm\; d\}\backslash xi\; \backslash frac\{1\}\{2\backslash sqrt\{\backslash pi\backslash Delta\; t\}\}\{\backslash rm\; e\}^\{-\backslash frac\{\backslash xi^2\}\{4\backslash Delta\; t\}\}\{\backslash rm\; e\}^\{-\{\backslash rm\; i\}k\backslash xi\}$$

The second part is a Gaussian Integral, which yields

$$P\; =\; \backslash int\; \backslash frac\{\{\backslash rm\; d\}k\}\{2\backslash pi\}\{\backslash rm\; e\}^\{\{\backslash rm\; i\}k(\{X\}\_\{n+1\}-\{X\}\_n\; +\; \backslash nabla\; U(\{X\})\backslash Delta\; t)\}\{\backslash rm\; e\}^\{-k^2\backslash Delta\; t\}$$

Now consider the probability from initial $X\_0$

to final $X\_n$.

$$P(\backslash mathbf\{X\}\_n|\backslash mathbf\{X\}\_0)\; =\; \backslash int\; \backslash frac\{1\}\{2\backslash pi\}\backslash prod\_i^\{N-1\}\; \{\backslash rm\; d\}k\_i\; \{\backslash rm\; e\}^\{(\{\backslash rm\; i\}k\_i(\backslash dot\{X\}\; +\; \backslash nabla\; U(X))-k\_i^2)\backslash Delta\; t\}$$

take the limit of $\backslash Delta\; t\backslash to\; 0$,we will get

$$P(\backslash mathbf\{X\}\_n|\backslash mathbf\{X\}\_0)\; =\; \backslash int\; \backslash mathcal\{D\}[k]\; \{\backslash rm\; e\}^\{\backslash int\_0^\{t\_n\}(\{\backslash rm\; i\}k(\backslash dot\{X\}\; +\; \backslash nabla\; U(X))-k^2)\{\backslash rm\; d\}\; t\}$$

• Hamiltonian mechanics
• Statistical mechanics
• Implicit solvation
• Stochastic differential equations
• Langevin equation
• Langevin Monte Carlo
• Klein–Kramers equation

{{reflist}}

• [http://cmm.cit.nih.gov/intro_simulation/node24.html Langevin Dynamics (LD) Simulation]

Category:Classical mechanics

Category:Statistical mechanics

Category:Dynamical systems

Category:Symplectic geometry