The generalized Langevin equation

The solution of the many-body problem as defined in the previous section on classical mechanics, is numerically demanding. Considering, furthermore, that the primary effect of the surface atoms is assumed to be that of a heat bath maintained at a certain temperature, it is naturally to think that some approximate description would suffice. 

In order to formulate a theory and derive the equations more generally, we divide the N atoms of the solid in two groups consisting of P primary atoms (see Fig. 7.1) and N — P secondary ones. From the hamiltonian (6.12) we get the equations of motion 

FfQURE 7.1 The atom A Interacts with the primary atoms (P). A generalized Langevin equation can be derived for the dynamics of the P surface atoms. 

The proof of these expression may easily be found by substituting in eqs. (7.4), and (7.5). In eq. (7.9) L(t) is a memory function picking up values of the solution in the range (0 t t) whereas F(r) appears as a fluctuating random force. Eq. (7.9) is only useful if some approximate scheme for the memory function L(r) and the fluctuating force F(t) can be introduced. A memory function already appeared in section 2.3 when energy transfer to the linear chain was considered. 

An important theorem known as the fluctuation dissipation theorem [86] connects the dissipative force L(/) to the ensemble average over the fluctuations through the equation 

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). 

The Markovian picture cannot be used to describe such motions. The generalized Langevin equation 

It is important to point out that this does not imply that Markovian stochastic equations cannot be used in descriptions of condensed phase molecular processes. On the contrary, such equations are often applied successfully. The recipe for a successful application is to be aware of what can and what cannot be described with such approach. Recall that stochastic dynamics emerge when seeking coarsegrained or reduced descriptions of physical processes. The message from the timescales comparison made above is that Markovian descriptions are valid for molecular processes that are slow relative to environmental relaxation rates. Thus, with Markovian equations of motion we cannot describe molecular nuclear motions in detail, because vibrational periods (10 s) are short relative to environmental relaxation rates, but we should be able to describe vibrational relaxation processes that are often much slower, as is shown in Section 8.3.3. 

Coming back to the non-Markovian equations (8.61) and (8.62), and their Markovian limiting form obtained when Z(Z) satisfies Eq. (8.60), we next seek to quantify the properties of the thermal environment that will determine its Markovian or non-Markovian nature. 

Note the difference between Eqs (8.65) and (6.90) or (7.79). The mass nij appears explicitly in (8.65) because here we did not use mass weighted normal mode coordinates as we did in Chapters 6 and 7. In practice this is just a redefinition of the coupling coefficient Cj. 

The assumption that f(t) is purely random may actually be a highly restrictive and unnecessary limitation. In fact, many physical processes cannot be described by Equation 1.4 simply because they exhibit memory effects. Thus, we must introduce an extended Ornstein-Uhlenbeck process, deflned as the solution of the most general linear stochastic equation with additive noise, which we write as 

In fact, the demonstration that the stationary condition leads to such a general fluctuation-dissipation relation also leads to very stringent and rigid conditions, of purely mathematical nature, on the structure of the relaxation matrix H(t). Thus, the so-called theoran of stationarity, states [22] that the stationarity condition alone is in fact equivalent to the condition that Equation 1.7 must be such that it can be formatted as the following general equation. 

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Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Second, the classical dynamics of this model is governed by the generalized Langevin equation of motion in the adiabatic barrier [Zwanzig 1973 Hanggi et al. 1990 Schmid 1983],... 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Cortes, E., West, B. J. and Lindenberg, K. On the generalized Langevin equation classical and quantum mechanical, J.Chem.Phys., 82 (1985), 2708-2717... 

In Kramers classical one dimensional model, a particle (with mass m) is subjected to a potential force, a frictional force and a related random force. The classical equation of motion of the particle is the Generalized Langevin Equation (GLE) ... 

An alternative view of the same physical process is to model the interaction of the reaction coordinate with the environment as a stochastic process through the generalized Langevin equation (GEE)... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Use of this v (t) as the friction (t) in the generalized Langevin equation provides a complete specification of a nonlocal stationary stochastic dynamics with the exponential friction jo. 

Exercise. The Ornstein-Uhlenbeck process (IV.3.10), (IV.3.11) satisfies the generalized Langevin equation with memory kernel ... 

It was also shown that the generalized Langevin equation is an exact equation of motion for v provided y(t) is the exact memory function, K (t).3SA2... 

In view of the fact that the correlation function for the random force, as given by Eq. [16], is a Dirac 8 function, the strict Langevin equation (Eq. [15]) is not amenable to computer simulation. In order to circumvent the above difficulty, it is convenient to describe the motion of the fictitious particle by the generalized Langevin equation. The generalized Langevin equation, which can be derived from the Liouville equation (11), along with the supplementary conditions are... 

The simple friction coefficient f in Eq. [15a] is replaced in the generalized Langevin equation (Eq. [21]) by a friction kernel containing the memory function /(r). In addition, in place of the condition given by Eq. [16], the second fluctuation-dissipation theorem provides the more general condition (11)... 

The Langevin equation [15a] is a special case of the generalized Langevin equation [21] when the memory function is given by... 

The particle trajectories can be simulated using a random force in the generalized Langevin equation that is constant during a small time step ts with values given by a Gaussian distribution. The memory function for this form of random force is (12)... 

In order to simulate the motion of the fictitious particle by an equation which constitutes a close approximation of the strict Langevin equation [15], we choose in the generalized Langevin equation ts 4 tp (=l/ )... 

On the other hand, if the position r(n) of the particle is smaller than the decay length of the interaction potential, the external foroe experienced by the particle during the time interval t < t < cannot be assumed constant because the interaction potential is a strong function of position. In order to discretize the generalized Langevin equation, the time interval t < t < t +i is further subdivided into K subintervals such that the... 

As will be seen below, the kernel y(f) decreases on a characteristic time of the order of mf. 1, the angular frequency ooc characterizing the bandwidth of the bath oscillators effectively coupled to the particle. The quantity y(t — t,) is thus negligible if (i)c(t - U) 1. Mathematically, this condition can be realized at any time t by referring the initial time f, to -oo. Then, the initial particle position becomes irrelevant in Eq. (10), which takes the form of the generalized Langevin equation, namely,... 

We now turn to quantum Brownian motion as described by the generalized Langevin equation (22), with the symmetrized correlation function of the random force as given by Eq. (20). 

As stated above, the Langevin force F(t) can be viewed as corresponding to a stationary random process. Clearly, the same is true of the solution v(f) of the generalized Langevin equation (22), an equation which is valid once the limit ti —> —oo has been taken. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used to obtain the velocity correlation function, which only depends on the observation time Cvv(t, t2) = Cvv(t —12). As in the classical case, the velocity does not age. 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

The validity of the generalized Langevin equation (22) is restricted to a stationary medium. In other situations, for instance when the diffusing particle evolves in an aging medium such as a glassy colloidal suspension of Laponite [8,12,55,56], another equation of motion has to be used. 

Let us consider again the particular case of a particle diffusing in a stationary medium, in order to see how the generalized Langevin equation (22) can be deduced from the more general equation (169). When the medium is stationary, the response function x (M0 reduces to a function of t — t (% (t, t ) = X (f — t j). Introducing then the causal function y(f) as defined by... 

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