We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.
We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical
d-dimensional system is mapped onto the statics of an associated quantum system and of its classical (
d+1)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.
Abstract Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.
General discussion of stochastic models In the hydrodynamic theories, one writes equations which describe the system at hand on long time and length scales, using local conservation laws and the presence of broken symmetries to determine the identity of these slowly varying quantities. However, one would like a way to take account of the effects of the more rapidly varying degrees of freedom in such theories. Implicitly, these more rapid degrees of freedom are present in the integrals which determine the transport coefficients in the hydrodynamic equations by way of Kubo relations, but no method is presented to calculate the required integrands. One way to take account of the other degrees of freedom is to take on the entire many body problem all the way down to the atomic or electronic level, as one does in molecular dynamics simulations of various sorts. However, it is useful to have some approximate analytical ways to attack this problem as well. Here we review the basis for the most common such approach, the Langevin equation. For most of this discussion, we will assume that the identity of the slow variable or variables is known and that the separation of time scales is extreme so that the faster variables are essentially instantaneous in a sense we will discuss. These assumptions are not often particularly well justified. However, the resulting formulation has yielded very useful insights and is an important part of the subject.
The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-
n limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.
We study the effect of quenched time-independent random fields coupled linearly to the order parameter on the dynamical critical behavior of spin systems. We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown that the dominant fluctuations are those induced by the random fields and, therefore, thermal fluctuations are irrelevant. This allows us to establish a relation between this model and a quantum spin system in the presence of a quenched random field. Moreover, we find that only static exponents in
D dimensions are the same as those of the pure ($D\ensuremath{-}2$)-dimensional theory, but the dynamical exponent
z does not satisfy this relation. The quantum system in
D dimensions is studied through its (
D+1)-dimensional equivalent model where the quenched random fields are totally correlated in the additional imaginary-time ($\ensuremath{\tau}$) direction. The system is anisotropic, and there is a new exponent
zA associated with the scaling behavior in the $\ensuremath{\tau}$ direction. We find the relation
z=2zA to all orders in perturbation theory. For the zero-temperature quantum model we find that the static (zero-frequency) exponents are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. At finite temperature, when the classical system is finite in the $\ensuremath{\tau}$ direction, we predict a crossover to
D-dimensional classical behavior in nonstatic response and correlation functions, with crossover exponent ${({z}_{A}{\ensuremath{\nu}}_{(D\ensuremath{-}2)})}^{\ensuremath{-}1}$, where ${\ensuremath{\nu}}_{(D\ensuremath{-}2)}$ is the exponent $\ensuremath{\nu}$ for the ($D\ensuremath{-}2$)-dimensional pure classical system. The static correlation functions do not have this cross-over behavior and are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. The dimensional shift in static quantities for both quantum and classical dynamical systems is a consequence of a supersymmetry in the underlying field theory. The exponent ${z}_{A}=1+\ensuremath{\eta}+O(\frac{1}{{N}^{2}})$ in the large-
N limit or ${z}_{A}=1+c\ensuremath{\eta}+O({\ensuremath{\epsilon}}^{4})$ in the $\ensuremath{\epsilon}$ expansion where $c=1\ensuremath{-}\frac{3}{4}[\frac{(N+2)}{{(N+8)}^{2}}]\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is the same as in the ($D\ensuremath{-}2$)-dimensional pure classical system. We also study the above classical random-field Ising problem using the interface approach, but are unable to draw any definite conclusion about the dynamics at the lower critical dimension
D=3.