On the origin of thermality
  It is well-known that a small system weakly coupled to a large energy bath in a total microcanonical ensemble will find itself in an (approximately) thermal state and, recently, it has been shown that, if the total state is, instead, a random pure state with energy in a narrow range, then the small system will still be approximately thermal with a high probability (wrt `Haar measure'). We ask what conditions are required for something resembling these 'traditional' and 'modern' thermality results to still hold when system and energy bath are of comparable size. In Part 1, we show that, for given system and energy-bath densities of states, s_S(e) and s_B(e), thermality does not hold in general, as we illustrate when both increase as powers of energy, but that it does hold in certain approximate senses, in both traditional and modern frameworks, when both grow as exp(be) or as exp(qe^2) and we calculate the system entropy in these cases. In their 'modern' version, our results rely on new quantities, which we introduce and call the S and B 'modapprox' density operators, which, we claim, will, with high probability, give a close approximation to the reduced density operator for the system and energy bath when the total state of system plus energy bath is a random pure state with energy in a narrow range. In Part 2 we clarify the meaning of these modapprox density operators and give arguments for our claim.   The prime examples of non-small thermal systems are quantum black holes. Here and in two companion papers, we argue that current string-theoretic derivations of black hole entropy and thermal properties are incomplete and, on the question of information loss, inconclusive. However, we argue that these deficiencies are remedied with a modified scenario which relies on the modern strand of our methods and results here and is based on our previous 'matter-gravity entanglement hypothesis'. 
Bernard S. Kay (York)
cond-mat.stat-mech gr-qc hep-th math-ph math.MP quant-ph