Quantum Effects in Neural Networks

We develop the statistical mechanics of the Hopfield model in a transverse field to investigate how quantum fluctuations affect the macroscopic behavior of neural networks. When the number of embedded patterns is finite, the Trotter decomposition reduces the problem to that of a random Ising model. It turns out that the effects of quantum fluctuations on macroscopic variables play the same roles as those of thermal fluctuations. For an extensive number of embedded patterns, we apply the replica method to the Trotter-decomposed system. The result is summarized as a ground-state phase diagram drawn in terms of the number of patterns per site, $\alpha$, and the strength of the transverse field, $\Delta$. The phase diagram coincides very accurately with that of the conventional classical Hopfield model if we replace the temperature T in the latter model by $\Delta$. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.Comment: 34 pages, LaTeX, 9 PS figures, uses jpsj.st

PRE-VITRIFICATION VISCOSITY ENHANCEMENT BY LONGITUDINAL MODE COUPLING

On propose un mécanisme par lequel la viscosité d'un liquide simple s'accroît suivant η ∝ (T-T0)-1 près d'une valeur finie de T0. Ce mécanisme comprend deux étapes : 1) un ralentissement de la relaxation structurelle contrôlée par la diffusion, au cas de η → ∞ ; 2) une amplification de la viscosité par suite du couplage mode-mode aux modes longitudinaux qui relaxent lentement à cause de 1).A mechanism is described for the viscosity of a simple liquid to grow as η ∝ (T-T0)-1 at some finite T0. It consists of two main steps : 1) a slowing-down of diffusion-controlled structural relaxation of density fluctuations as η → ∞ ; 2) an enhancement of viscosity by mode-mode coupling to longitudinal modes which are slowly relaxing in view of 1)