Markovian random forces

A chemical reaction, for instance, an intramolecular reaction R P, is viewed as a one-dimensional motion in the phase space of a particle in a double-well potential and undergoing Markovian random forces. The dynamics of the particle are described in terms of the Kramers equation (4.160), and the rate of reaction can, in principle, be calculated from the knowledge of the probability density function p(x0 t, x).16,147... 

The important point is that the form of the Langevin equation in Eq. (2.5) is preserved under any canonical point transformation. For the case of Markovian random forces (constant friction),... 

The Kramers model consists of a classical particle of mass m moving on a one-dimensional potential surface V(x) (Fig. 1) under the influence of Markovian random force R(t) and damping y, which are related to each other and to the temperature T by the fluctuation dissipation theorem. 

When the solvent is treated as a continuous dielectric background that interacts stochastically with the mobile ions, the ionic trajectories can be modeled with the Langevin formalism. " In particular, the strict or full Lange-vin equation can be used, which assumes Markovian random forces and neglects correlations (both spatially and temporally) of the ionic motion ... 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

We have already argued (Section 7.4.2) that the Markovian nature of the system evolution implies that the relaxation dynamics of the bath is much faster than that of the system. The bath loses its memory on the timescale of interest for the system dynamics. Still the timescale forthe bath motion is not unimportant. If, for example, the sign of Rf) changes infinitely fast, it makes no effect on the system. Indeed, in order for a finite force R to move the particle it has to have a finite duration. It is convenient to introduce a timescale tb, which characterizes the bath motion, and to consider an approximate picture in which Rf) is constant in the interval [t, t -I- Tb], while Rff and Rff are independent Gaussian random variables if... 

In a previous section reference was made to the random walk problem (Montroll and Schlesinger [1984], Weiss and Rubin [1983]) and its application to diffusion in solids. Implicit in these methods are the assnmptions that particles hop with a fixed jump distance (for example between neighboring sites on a lattice) and, less obviously, that jumps take place at fixed equal intervals of time (discrete time random walks). In addition, the processes are Markovian, that is the particles are without memory the probability of a given jump is independent of the previous history of the particle. These assumptions force normal or Gaussian diffusion. Thus, the diffusion coefficient and conductivity are independent of time. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

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