# Noncommutative Geometry Seminar

*,*McGill University

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We study the first law of thermodynamics in the setting of a finite dimensional quantum system coupled to an infinitely extended thermal reservoir. Using a two-time measurement protocol, in which measurements of the system and reservoir energy are made at time 0 and at some later time t, the first law states that in the limit of large t and small coupling energy, the change of energy in the reservoir should be equal to the negative change of energy of the system. It is well-known that this holds in the sense of expected values of the energy measurements. We define measures which encode the full statistics of the energy changes, and show that the measures themselves converge weakly, which is significantly stronger. To do this, we write the measures in terms of a relative modular operator and exploit the machinery of modular theory.