In this paper, we address an important question of the relationship between fluctuation theorems for the dissipated work W d = W −∆F with general finitetime (like Jarzynski equality and Crooks relation) and infinite-time (like Gallavotti-Cohen theorem) drive protocols and their time-reversal symmetric versions. The relations between these kinds of fluctuation relations are uncovered based on the examples of a classical Markovian N -level system. Further consequences of these relations are discussed with respect to the possible experimental verifications.

We introduce a new toy model for the study of glasses: the hardmatrix model (HMM). This may be viewed as a single particle moving on SO(N ), where there is a potential proportional to the 1-norm of the matrix. The ground states of the model are "crystals" where all matrix elements have the same magnitude. These are the Hadamard matrices when N is divisible by four. Just as finding the latter has been a difficult challenge for mathematicians, our model fails to find them upon cooling and instead shows all the behaviors that characterize physical glasses. With simulations we have located the first-order crystallization temperature, the Kauzmann temperature where the glass has the same entropy as the crystal, as well as the standard, measurement-time dependent glass transition temperature. A new feature in this glass model is a "disorder parameter" ρ 0 that is zero for any of the crystal phases and in the liquid/glass, where it is non-zero, corresponds to the density of matrix elements at the maximum in their contribution to the energy. We conclude with speculation on how a quantum extension of the HMM, with the backdrop of current work on many-body localization, might advance the understanding of glassy dynamics.Much of statistical mechanics is the study of "toy models," minimalistic distillations of physical systems that capture particular phenomena. The simplest model of liquids, mono-disperse hard spheres, is also much used as a model of glassy behavior. In three dimensions, and when compressed rapidly, this system produces jammed structures with a reproducible packing

We introduce a new toy model for the study of glasses: the hard-matrix model. This may be viewed as a single particle moving on SO(N), where there is a potential proportional to the one-norm of the matrix. The ground states of the model are ‘crystals’ where all matrix elements have the same magnitude. These are the Hadamard matrices when N is divisible by four. Just as finding the latter has challenged mathematicians, our model fails to find them upon cooling and instead shows all the behaviors that characterize physical glasses. With simulations we have located the first-order crystallization temperature, the Kauzmann temperature where the glass would have the same entropy as the crystal, as well as the standard, measurement-time dependent glass transition temperature. Our model also brings to light a new kind of elementary excitation special to the glass phase: the rubicon. In our model these are associated with the finite density of matrix elements near zero, the maximum in their contribution to the energy. Rubicons enable the system to cross between basins without thermal activation, a possibility not much discussed in the standard landscape picture. We use these modes to explain the slow dynamics in our model and speculate about their role in its quantum extension in the context of many-body localization.