The Fokker-planck Operator at a Continuous Phase Transition

I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The 4 M theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry etc. The eigenfunctions of the corresponding Fokker-Planck operator can be chosen, of course, to be eigenfunctions of the momentum operator. It is shown that the eigenvalues of the FP operator, corresponding to each eigenvalue q of the momentum operator, evaluated at a transition point of the finite system, accumulate at zero, when the size of the system tends to infinity. There are many reasonable ways of defining a critical temperature of a finite system, that tends to the critical temperature of the infinite system as the size of the system tends to infinity. The accumulation of eigenvalues is neither affected by the specific choice of critical temperature of the finite system nor by whether the system is below or above its upper critical dimension. The property of critical slowing down [1,2] is known for a long time from experimental [3-7] and numerical work [8-11]. The quantitative description is in terms of characteristic decay times or alternatively " characteristic frequencies " q Z , that govern the decay of a disturbance of wave vector q. It was found that at the transition, q Z behaves as some positive power of q , z q q v Z. This implies that the larger the scale of the disturbance the longer it takes to decay and a divergent scale results in a divergent decay time. The evaluation of the exponent z was the subject of theoretical work using a number of different approaches but all based on stochastic field equations of the Langevin type to describe the dynamics of the system [12-18]. More recently, it was suggested that not only do the characteristic frequencies tend to zero with q but the decay function itself becomes slower than an exponential (e.g. stretched exponential) for long times, 1 !! t q Z [19-21]. That property was shown to hold for quite general Langevin field equations including in addition to critical dynamics, equations of the KPZ type. The derivation of stretched exponential decay or any other form of slow …