Equilibrium states at ¿negative¿ absolute temperature are found when the entropy of a system is a decreasing function of its total energy, i.e. when providing additional energy results in a contraction of the phase-space volume accessible to the system. Negative absolute temperatures are well known to characterize, for instance, the high-energy limit of nuclear spins systems, cold atoms trapped in optical lattices and interacting vortexes in two-dimensional hydrodynamics.
In recent years studies on negative-temperature equilibrium states were also carried out by mean of simple one-dimensional chains of Hamiltonian oscillators with bounded and periodic kinetic terms. The aim of this project is to study the behavior of those chains when they are driven out of equilibrium by two thermal baths imposing different temperatures at the boundaries (even with opposite signs). A systematic investigation of those models will allow for a better understanding of dynamical properties of systems with negative temperature in out-of-equilibrium conditions.
An extensive employment of GPU computer simulations will be able to characterize the properties of out-of-equilibrium response functions in those models, in presence of negative absolute temperature conditions. This will be an important step to understand the relation between response, temperature and transport properties. A particularly intriguing situation is obtained by imposing temperatures with opposite signs to the boundaries of the chain, resulting in a spatial cross-over between positive and negative temperature; in this case it is not clear what would be the effect on the T>0 region of a local perturbation in the T
The study of linear response theory in systems with negative temperature is a quite new topic. In a previous work [1] the case of an equilibrium chain with long-range interactions has been considered, but to the best of my knowledge it is the only case. Statistical features of systems with negative temperature out of equilibrium have also been studied in very few cases. Some works have considered the Discretized Non-Linear Schroedinger (DNLS) equation, an Hamiltonian system whose behavior can be studied exactly only at equilibrium. This system can show negative absolute temperature, and numerical simulations have been done to understand its out-of-equilibrium properties [2-4]. However, to the best of my knowledge, no systematic study of response functions has never been attempted in out-of-equilibrium conditions in presence of local negative temperature.
The project focuses on the study of an Harmonic chain living in a bounded phase-space and thus allowing for negative temperature. Due to the form of the Hamiltonian, it is possible to describe the average evolution of some suitable dynamical observables as functions of time. This point is of particular interest, since it has been shown [2-4] that the DNLS equation shows transitory non-equilibrium negative-temperature states even when it is coupled to thermal baths at positive temperatures. It would be of great interest to detect similar behavior in the simple Hamiltonian system under study, in particular if they could be foreseen by analytical predictions.
The proposed study should also bring some insight on the relation between response functions, correlations and temperature in an out-of-equilibrium regime presenting a temperature profile. In particular, the effect of the change of temperature sign along the chain on the response functions is not completely clear and its numerical investigation would shed some light on this intriguing aspect. We could, for instance, impose temperatures with opposite signs to the boundary of the chain, resulting in a temperature profile along the lattice, passing from a regime of locally negative temperature to a regime with T>0; in this peculiar out-of-equilibrium situation, one could study the effect on the T>0 zone after a local perturbation in the T
Under a wider point of view, the proposed research topic could have some impact because of its innovative elements in the study of Fourier law. The study of heat transport along one-dimensional systems is a very active research field, and a lot of different models have been proposed and analyzed during the years in order to define and isolate universal properties. The present model is not only a suitable candidate for a nonlinear chain, whose typical behavior should be compared to those already investigated; it is also a simple example in which heat could be transferred among baths with temperature of opposite signs, (partially) solving some of the controversial aspects in the usage of negative temperature to describe the statistical mechanics of bounded systems [5-7].
[1] F. Miceli, M. Baldovin, A. Vulpiani, Phys. Rev. E 99, 042152 (2019).
[2] S. Iubini, S. Lepri, and A. Politi, Phys. Rev. E 86, 011108 (2012).
[3] S. Iubini et al., New J. Phys. 15, 023032 (2013).
[4] S. Iubini, A. Politi, and P. Politi, J. Stat. Mech. 2017, 073201 (2017).
[5] J. Dunkel and S. Hilbert, Nature Physics 10, 67-72 (2014).
[6] S. Hilbert, P. Hänggi, and J. Dunkel, Phys. Rev. E 90, 062116 (2014).
[7] H. Struchtrup Phys. Rev. Lett. 120, 250602 (2018).