Abstract
AbstractWe prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.
Highlights
Equipartition of energy is a general principle in classical statistical physics stating that in an ergodic system at equilibrium, the total energy is shared among the elementary degrees of freedom
In some special cases this principle could be verified; see [4] and references therein for an extensive physics literature on the popular model of a single quantum particle in contact with a quantum heat bath consisting of infinitely many harmonic oscillators
In this paper we show that for Wigner random matrices – that is, for a mean-field quantum system with random quantum transition rates – a strong microcanonical form of the quantum equipartition holds: it is valid separately for every eigenvector
Summary
Equipartition of energy is a general principle in classical statistical physics stating that in an ergodic system at equilibrium, the total energy is shared among the elementary degrees of freedom. Fine properties of eigenvectors of large Wigner matrices have been extensively studied in recent years They are delocalised – that is, max | ( )| ≤ −1/2+ for any fixed > 0, with very high probability as tends to infinity. The eigenvalues are indexed in an increasing order, 1 ≤ 2 ≤ · · · ≤ In all these previous results the eigenvector was tested against a specific deterministic observable; but in the equipartition relation (1.1) we consider the quadratic form of with a random that is far from being independent of. Given the complicated dependence between and , it is somewhat surprising that the proof of formula (1.1) is simpler than those of formulas (1.2) and (1.3) Despite this dependence, we can still directly handle ( , ) for an individual eigenvector – that is, we do not need to first establish a spectrally local-averaged version of formula (1.1) in the form.
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