Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

Abstract

Nonlinear generalized Langevin equation with memory due to thermal environment is equivalent to a memoryless equation with increased dimensionality A framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in multidimensional Langevin equation is extended to the generalized Langevin equations without a priori assumption on the forms of the potential (in general, nonlinearly coupled systems) and the friction kernel. The equation of motion with memory effect can be transformed into an equation without memory at the cost of an increase in the dimensionality of the system, and hence the theoretical framework developed for the (nonlinear) Langevin formulation can be generalized to the non-Markovian process with colored noise. It is found that the increased dimension can be physically interpreted as effective modes of the fluctuating environment. As an illustrative example, we apply this theory to a multidimensional generalized Langevin equation for motion on the Müller-Brown potential surface with an exponential friction kernel. Numerical simulations find a boundary between the highly reactive region and the less reactive region in the space of initial conditions. The location of the boundary is found to depend significantly on both the memory kernel and the nonlinear couplings. The theory extracts a reaction coordinate whose sign determines the fate of the reaction taking into account the thermally fluctuating environments, the memory effect, and the nonlinearities. It is found that the location of the boundary of reactivity is satisfactorily reproduced as the zero of the statistical average of the new reaction coordinate, which is an analytical functional of both the original position coordinates and velocities of the system, and of the properties of the environment.

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.…”

“…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).…”

Why and how do systems react in thermally fluctuating environments?

“…Recent developments in reaction dynamics theories under the existence of thermally fluctuating environments 5,52,[64][65][66][67][68][76][77][78][79] have made great progress in understanding the origins of reactions, that is, what type of initial conditions can bring the system to the reactant or the product. It was shown 5,[64][65][66][67][68] that, by introducing a coordinate shift for a given realization (i.e., time sequence) of the random force, the phase space structure dividing reactive and nonreactive trajectories can be extracted exactly in the case of a quadratic barrier without any nonlinear couplings among the modes.…”

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.…”