While classical and quantum regimes are typically regarded as distinct, classical dynamics can, in fact, be used to determine the time evolution of a quantum many-body system under non-equilibrium, thus doing away with a full-blown quantum machinery. Essentially, the classical equation of motion for the phase space distribution can provide an approximate solution of the quantum equation of motion for the density matrix (matrix describing a quantum system’s statistical state), if the initial classical analogue of the quantum distribution function is known. This beneficial feature of classical dynamics can even be extended to the case of field theories, where, instead of discrete degrees of freedom, we deal with continuous degrees of freedom.

However, there is a caveat: the classical dynamics approach is less useful when describing equilibrium behavior of quantum systems. For instance, in the case of black-body cavity radiation, classical considerations lead to the Rayleigh-Jeans divergence. Thus, a framework that can help us reach quantum statistical equilibrium following a classical dynamical evolution is highly desired.

To that end, the authors propose a novel framework called “the replica evolution method” in which quantum statistics is realized by solving classical equations of motion. To start with, they consider a set of classical field configurations (Φ_{xτ},π_{xτ}_{}) in (3+1) dimensions (three spatial dimensions, one replica index (τ) dimension), called “replicas,” where each replica interacts with its nearest neighbors through a τ-derivative term that eventually leads to their thermalization. The time evolution of the replicas is governed by the classical Hamilton’s equation of motion.

The most interesting feature of their framework is that the replica index ‘τ’ can be regarded as the imaginary time index in finite temperature quantum field theory, so that the replica evolution is technically similar to the fictitious time evolution in a hybrid Monte Carlo simulation. This, in turn, implies that, given a distribution of initial replica Hamiltonian values, the replicas should relax to the equilibrium quantum distributions following a long-time evolution. Moreover, the τ-averaged field variables obey classical equations of motion when the fluctuations among replicas are small. From these two aspects, one can expect that replica evolution would describe the real time evolution of quantum systems. Actually, the time-correlation function in the replica evolution is found to reproduce that in quantum field theory to a good approximation. This framework can thus allow for more precise predictions of relaxation processes in quantum systems and possibly help unify the background field and the particle degrees of freedom, paving the way for an integrated transport theory.

]]>Mathematicians and physicists are well acquainted with second-order ordinary differential equations (ODE), the most prominent of them being the class of equations that govern oscillatory motion. Often, these oscillatory-type equations involve an inhomogeneous “potential” term coupled with a perturbation term and are usually solved as a power series in the perturbation with coefficients that are polynomials in the independent variable. However, when describing time-dependent behavior of a system, i.e., the independent variable is time (t), the series solution becomes problematic due to the coefficients diverging in time, which limits the valid description of the system to time scales of the order of (1/perturbation).

In a new study, I report on an elegant (yet unnoticed) functional identity in renormalization group (RG) theoretical perturbation theory, which is useful to get rid of the divergent time-dependence and to replace it instead with a description based on “renormalized amplitudes” and their time dynamics.

I begin with a second-order ODE in time with a potential term depending on the perturbation, the independent variable and its derivative, and the “resonant harmonics,” e^{±it}, and its naïve perturbation series solution. Next, I write the dependent variable, say y, as a perturbation series expansion and substitute it in the ODE to get an equation for each power of the perturbation. I define Y as the resulting formal solution of the power series expansion of y and write it as a Fourier expansion with coefficients that are functions of what I call “bare amplitudes,” A, B. I will formally refer to them as “secular coefficients.”

I go on to prove an identity and a relation between the secular coefficients that are key to my results. I next define the “renormalized amplitudes” A_{r}(t) and B_{r}(t) as a function of the “resonant” secular coefficient and use the proved relation to express the bare amplitudes as a function of A_{r}(t) and B_{r}(t). Using the identity proved before, I further establish the renormalized series expansion in terms of A_{r }and B_{r}, which is free from the secular time dependence.

Further, the dynamics of A_{r}(t) and B_{r}(t) is solely determined by the resonant secular coefficients to all orders of perturbation.

This work has important implications in studies involving the classic cases of Van der Pol, Mathieu, Duffing, and Rayleigh equations and may trigger new theoretical development of a RG-based approach to solving differential equations.