Z Skrypty dla studentów Ekonofizyki UPGOW

Use in Physics

In physics, SDEs are typically written in the Langevin form and referred to as "the Langevin equation." For example, a general coupled set of first-order SDEs is often written in the form:

\[\dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum_{m=1}^ng_i^m(\mathbf{x})\eta_m(t),\,\]

where \(\mathbf{x}=\{x_i|1\le i\le k\}\) is the set of unknowns, the \(f_i\) and \(g_i\) are arbitrary functions and the \(\eta_m\) are random functions of time, often referred to as "noise terms". This form is usually usable because there are standard techniques for transforming a higher-order equations into several coupled first-order equations by introducing new unknowns. If the \(g_i\) are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case where \[ g(x) \propto x\]. Additive noise is the simpler of the two cases; in that situation the correct solution can often be found using ordinary calculus and in particular the ordinary chain rule of calculus. However, in the case of multiplicative noise, the Langevin equation is not a well-defined entity on its own, and it must be specified whether the Langevin equation should be interpreted as an Ito SDE or a Stratonovich SDE.

In physics, the main method of solution is to find the probability distribution function as a function of time using the equivalent Fokker-Planck equation (FPE). The Fokker-Planck equation is a deterministic partial differential equation. It tells how the probability distribution function evolves in time similarly to how the Schrödinger equation gives the time evolution of the quantum wave function or the diffusion equation gives the time evolution of chemical concentration. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques include the path integration that draws on the analogy between statistical physics and quantum mechanics (for example, the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables) or by writing down ordinary differential equations for the statistical moments of the probability distribution function.

Note on "the Langevin equation"

The "the" in "the Langevin equation" is somewhat ungrammatical nomenclature. Each individual physical model has its own Langevin equation. Perhaps, "a Langevin equation" or "the associated Langevin equation" would conform better with common English usage.