Energy master equation: A low-temperature approximation to Bässler’s random-walk model

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Abstract
The first part of this paper deals with the justification of Bässler’s phenomenological random-walk model for viscous liquids [Phys. Rev. Lett. 58, 767 (1987)], which considers the random walk of a ‘‘particle’’ representing the liquid state on a d-dimensional infinite cubic lattice with site energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton’s laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk model—the energy master equation (EME)—is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures at long times in high dimensions. However, computer simulations show that the EME works well already in two dimensions and at only moderately low temperatures. The EME has no randomness and no fitting parameters. The EME is completely specified from the density of states and the attempt frequency of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent with statistical mechanics. The final part of the paper gives a comprehensive discussion, comparing the EME to related work and listing the EME’s qualitatively correct predictions, its new predictions, and some ‘‘wrong’’ predictions, most of which go against the common picture of viscous liquids and the glass transition without violating experiments.