Langevin equation, linear

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

Following the general procedure of geometric TST, we start by discussing the linearized dynamics in relative coordinates. If the definition (41) is substituted into the linearized Langevin equation (13), it yields an equation of motion for the relative coordinate ... 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

An example for the predictive power of the TS trajectory is shown in Fig. 4. This figure shows a random instance of the TS trajectory (black) and a reactive trajectory (red) for the linearized Langevin equation (15) with N = 2 degrees of... 

In the presence of noise, the solution to the linear Langevin equation (11) is (see Appendix A, and also Eq. (3-5) of the article by Einstein in this proceedings). 

Acknowledgements This work was supported by the NSF under grant DMR-9312839. Appendix A - Solution to the linear Langevin equation. 

I. Linear differential equations in which only the inhomogeneous term is a random function, such as the Langevin equation. Such equations have been called additive and can be solved in principle. 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

Now, in order to decouple the time dependence of the stochastic variables governed by the Langevin equations (M.4), introduce new stochastic variables R(f) and R(f), which are linear combinations of Q(f) and v(f), in a way that is not far from that used in quantum mechanics, when passing from Q and P to a and at. This may be performed according to... 

In Section II, motivated by the fact that in typical experiments an aging system is not isolated, but coupled to an environment which acts as a source of dissipation, we recall the general features of the widely used Caldeira-Leggett model of dissipative classical or quantum systems. In this description, the system of interest is coupled linearly to an environment constituted by an infinite ensemble of harmonic oscillators in thermal equilibrium. The resulting equation of motion of the system can be derived exactly. It can be given, under suitable conditions, the form of a generalized classical or quantal Langevin equation. 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

Interestingly, due to the linearity of the generalized Langevin equation (22), the same effective temperature T,eff(( ) can consistently be used in the modified Nyquist formula linking the noise spectral density C/ /- ([Pg.313]

The last comer of our ideal triangle symbolizes the reduced model theory (RMl). The last few years have heard some debates on how to build up a modeling approach to molecular dynamics in the liquid state. These earlier attempts are based mainly on the generalized Langevin equation of Mori, which is linear in nature. In this book we shall illustrate a nonlinear version of this approach, the application of which implies wide use of both the AEP and CFP. 

The purpose of this section is to develop a systematic approach to evaluating higher-order corrections to the results discussed in Section II. Of course, this systematic approach can also be applied to physical systems different from that discussed in Section II. As a consequence of the linear nature of this system, these rules prove the exact equivalence between the Newtonian description of Eq. (2.1) and the standard Langevin equation provided that the assumption of time-scale separation is satisfied (see Section IV). 

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