China
Africa
Explore chapters and articles related to this topic
Interfacial Kinetics and Hopping Transitions
Published in Juan Bisquert, The Physics of Solar Energy Conversion, 2020
The principle of detailed balance is an extension of the fundamental microscopic time reversibility, according to which for each quantum transition the exact reversal is also possible. Therefore, in equilibrium the forward rate must equal the reverse rate of that process. However, the application of this principle to phenomenological rates of charge transfer and chemical reaction, that actually involve macroscopic fluxes instead of individual transition events, requires a considerable leap of generalization. When we consider a reaction rate between reactants and products, a thermal average of both the starting and final states is required. If the reaction is described by classical trajectories going from reactants to products, there are the time-reversed trajectories going from products to reactants. Averaging over an ensemble of initial states, the rate constant for the forward reaction is thereby related to that for the backward reaction and the ratio between the two rate constants is the equilibrium constant. In general, detailed balance condition imposes the constraint that in the equilibrium steady state, the flux of forward and backward transitions between a pair of states balance in detail.
The Boltzmann equation and the relaxation time approximation
Published in DAVID K. FERRY, Semiconductor Transport, 2016
In detailed balance, the two scattering probabilities differ by, for example, differences in the density of final stares (the second argument) in energy space and by the phonon factor difference berween emission and absorption processes. In fact, (3.46) encompasses four processes when phonons are involved. Carriers can leave by either emitting a phonon (and going to a state of lower energy) or by absorbing a phonon (and going to a state of higher energy). By the same token, they can scatter into the state of interest either by phonon emission from a state of higher energy, or by phonon absorption from a state of lower energy. In equilibrium, the processes connecting our primary state with each of the two sets of levels (of higher and lower energy) must balance. This balancing in equilibrium is referred to as detailed balance. Under this condition, the right-hand side of (3.43) vanishes in equilibrium.
Structure-Dynamic Approach of Nanoionics
Published in Klaus D. Sattler, 21st Century Nanoscience – A Handbook, 2020
The principle of detailed balance is strictly satisfied in the equilibrium state of a system. Charge and voltage on the ECb/{Xj} blocking heterojunction depend on a current density of G(t) = IM(t) generator, i.e., on the integral ∫ IM(t) dt. Therefore, a smallness of local current densities in {Xj} was chosen as the criterion of a weakness of external influence.
Trajectory reweighting for non-equilibrium steady states
Published in Molecular Physics, 2018
Patrick B. Warren, Rosalind J. Allen
The Boltzmann factor, which describes exactly the relative probability of microstates at equilibrium in systems whose dynamics obeys detailed balance, forms the cornerstone of a plethora of simulation methods in the physical sciences. For example, the seminal Metropolis–Hastings algorithm for Monte-Carlo simulation exploits the Boltzmann factor to generate a trajectory of configurations which sample the Gibbs–Boltzmann distribution [1]. Knowledge of the Boltzmann factor also makes possible a host of biased sampling methods, which allow efficient characterisation of rugged free energy landscapes comprising multiple free energy minima separated by barriers. In these methods, information on a target system of interest is obtained by simulating a reference system, whose microstate probabilities are biased to be different from the target system. The results are corrected for the bias by reweighting with, for example, a Boltzmann factor. The reference system is typically easier to sample than the target system. Thus in umbrella sampling [2], an external potential is used to coerce the reference system (or a sequence of such systems) to sample a free energy barrier. The basis of biased sampling schemes is the generic relation where is some quantity of interest (an order parameter for example), the brackets refer to an average over microstates for the reference system, the brackets refer to an average for the target system and W is a reweighting factor Here, the ratio is the relative probability of observing the microstate , where and are the steady-state (superscript ‘∞’) probability distributions for the reference and target systems, respectively. For systems whose dynamics obeys detailed balance this ratio is given analytically by the Boltzmann factor (up to an overall constant of proportionality). Thus, Equation (1) provides a way to compute averages over the target system from a simulation of the reference system: during the simulation, one simply tracks the quantity W and uses it to reweight the average of the quantity of interest Θ. The constant of proportionality does not need to be calculated since it cancels in Equation (1).