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The first law of thermodynamics states that the average total energy current between different reservoirs vanishes at large times. In this note we examine this fact at the level of the full statistics of two times measurement protocols also known as the Full Counting Statistics. Under very general conditions, we establish a tight form of the first law asserting that the fluctuations of the total energy current computed from the energy variation distribution are exponentially suppressed in the large time limit. We illustrate this general result using two examples: the Anderson impurity model and a 2D spin lattice model.
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This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans--Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in [Andrieux et al. New J. Phys. 2009]. We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in [Andrieux emph{et al.} New J. Phys. 2009], we recover the Fluctuation-Dissipation Theorem in the linear response theory, short of Green--Kubo relations. We illustrate the general theory on a number of canonical models.
We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauers Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauers bound.
Motivated by fractional quantum Hall effects, we introduce a universal space of statistics interpolating Bose-Einstein statistics and Fermi-Dirac statistics. We connect the interpolating statistics to umbral calculus and use it as a bridge to study the interpolation statistics by the principle maximum entropy by deformed entropy functions. On the one hand this connection makes it possible to relate fractional quantum Hall effects to many different mathematical objects, including formal group laws, complex bordism theory, complex genera, operads, counting trees, spectral curves in Eynard-Orantin topological recursions, etc. On the other hand, this also suggests to reexamine umbral calculus from the point of view of quantum mechanics and statistical mechanics.
We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have been discussed for infinite matrix size. Moreover we derive the joint probability densities of the eigenvalues. To illustrate our results we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights we show that the product of complex random matrices yield a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.
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