Phase-space geometry of the generalized Langevin equation

Abstract

The generalized Langevin equation is widely used to model the influence of a heat bath upon a reactive system. This equation will here be studied from a geometric point of view. A dynamical phase space that represents all possible states of the system will be constructed, the generalized Langevin equation will be formally rewritten as a pair of coupled ordinary differential equations, and the fundamental geometric structures in phase space will be described. It will be shown that the phase space itself and its geometric structure depend critically on the preparation of the system: A system that is assumed to have been in existence for ever has a larger phase space with a simpler structure than a system that is prepared at a finite time. These differences persist even in the long-time limit, where one might expect the details of preparation to become irrelevant.

Derivation of the generalized Langevin equation in nonstationary environments

“…The other stimulating subject is the combination of the present theory and the recently developed dynamical reaction theory to extract the rigorous reaction coordinate to dominate the fate of reactions under thermal fluctuation in equilibrium. [6][7][8][9][10][11][12][13][14][15][16][17] These should provide us with great new insights into many molecular events occurring in nonstationary environments.…”

“…Reference 34 introduced a more generalized form of the friction kernel allowing for multiple heat baths and time dilatation in the arguments of the equilibrium friction kernels: (11) where k labels each heat bath and T k is the time-dependent "temperature" of the kth bath. The function τ k (t) is a monotonically increasing function of t with time-dependent rate of increase reflecting the change of the frequency (and therefore the characteristic response time) of each bath with time.…”

“…3 An example of such statistical property of the random force is the fluctuation-dissipation theorem, where the autocorrelation function of the random force is related to the friction kernel. It was found recently [6][7][8][9][10][11][12][13][14][15][16][17] that even though one cannot know an instantaneous value of the random force in advance since the initial condition of the bath is unknown, the statistical property enables us to analytically derive the boundary of the reaction in the state space, that is, a surface on which the system should end up with the reactant and the product with equal probability of one half. Following the pioneering works by Kramers 1 and by Grote and Hynes, 2 great progress [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] in the study of reaction dynamics in condensed phase have been made by using the GLE or the Langevin equation (a memoryless limit of GLE).…”

Rate calculation in two-dimensional barriers with colored noise

“…15) implies that the system was prepared in the remote past. It is crucial in order to make all of the six-dimensional phase [35]. In thermal equlibrium, 1 and 2 follow a Gaussian distribution with zero mean and correlations equal to…”

Phase space geometry of dynamics passing through saddle coupled with spatial rotation

“…The potential of the theories has been demonstrated not only in chemical reactions with 17,22 and without [23][24][25][26][27] time-dependent external field but also in ionization of a hydrogen atom in crossed electric and magnetic fields, [28][29][30] isomerization of clusters, [31][32][33][34][35][36] and the escape of asteroids from Mars 37,38 [Just recently the theory was also generalized to quantum Hamiltonian systems [39][40][41] and dissipative (generalized) Langevin systems. [42][43][44][45][46][47][48][49][50][51] The dimension of the phase space of an N -particle nonrigid system is (6N − 10) in the upper limit. 52 Nonrigid molecules at constant energy have ten constraints of the three coordinates of center of mass, the three conjugate momenta of center of mass, the three angular momenta (defined in the space-fixed frame), and the total energy of the system.…”