Rate, escape

The last term in (5.44) accounts for quantum corrections to the classical escape rate (5.31) [Wolynes 1981 Melnikov and Meshkov 1983 Grabert and Weiss 1984 Dakhnovskii and Ovchinnikov 1985]. 

C. Jaffe, S. D. Ross, M. W. Eo, J. Marsden, D. Farrelly, andT. Uzer, Statistical theory of asteroid escape rates, Phys. Rev. Lett. 89, 011101 (2002). 

The original work of Kramers [11] stimulated research devoted to calculation of escape rates in different systems driven by noise. Now the problem of calculating escape rates is known as Kramers problem [1,47]. 

Initially, an overdamped Brownian particle is located in the potential minimum, say somewhere between x and X2- Subjected to noise perturbations, the Brownian particle will, after some time, escape over the potential barrier of the height AT. It is necessary to obtain the mean decay time of metastable state [inverse of the mean decay time (escape time) is called the escape rate]. 

In the high barrier limit in this particular problem the inverse escape time is the Kramers escape rate. 

Let us consider a molecule absorbed on a metallic surface with an STM tip positioned at site k. This arrangement is schematically shown in Fig. la. One of the leads in this case is the tip, the other one is the metal itself. The escape rate matrix for the tip can be modeled as a local contact, only coupling the molecule at site k ... 

In this section, we consider a molecular junction [9-15]. The molecular chain is sandwiched between two metallic leads, and it is only coupled to them at its terminal sites. This arrangement is illustrated in Fig. Ih. The escape rate matrices now can he modeled as... 

Ab Initio Derivation of Entropy Production Escape-Rate Theory... 

B. Escape-Rate Formula in Dynamical Systems Theory... 

Proof with the Escape-Rate Theory Markov Chains and Information Theoretic Aspects Eluctuation Theorem for the Currents... 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The spontaneous breaking of time-reversal symmetry also manifests itself in the escape-rate theory, which consists in putting the system out of equilibrium by... 

The diffusive random walk of the Helfand moment is mled by a diffusion equation. If the phase-space region is defined by requiring Ga(t) < x/2, the escape rate can be computed as the leading eigenvalue of the diffusion equation with these absorbing boundary conditions for the Helfand moment [37, 39] ... 

Introducing these relations in Eq. (88) for the invariant probability, we obtain the escape-rate formula... 

SO that the escape rate can be directly related to the fractal dimension and the Lyapunov exponent ... 

If we combine the escape-rate formula (92) with the result (84) that the escape rate is proportional to the transport coefficient, we obtain the following large-deviation relationships between the transport coefficients and the characteristic quantities of chaos [37, 39] ... 

The formula (101) can also be proved with the escape-rate theory. We consider the escape of particles by difffusion from a large reservoir, as depicted in Fig. 17. The density of particles is uniform inside the reservoir and linear in the slab where diffusion takes place. The density decreases from the uniform value N/V of the reservoir down to zero at the exit where the particles escape. The width of the diffusive slab is equal to L so that the gradient is given by Vn = —N/ VL) and the particle current density J = —Wn = VN/ VL). Accordingly, the number of particles in the reservoir decreases at the rate... 

In the limit of an arbitrarily large reservoir N,V oo with n = N/V constant, the escape rate vanishes and a nonequilibrium steady state establishes itself in the diffusive slab. 

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

---