Abstract
The evolution dynamics of a large number of physical systems can be approximated by the discrete nonlinear Schrödinger equation (DNLSE). These include coupled spring-mass oscillators, Coupled pendulums, L-C electrical circuits, polymer chains, trapped ultra-cold atoms, optical waveguides, and more. Typically the DNLSE is applied to a slowly-varying envelope – a large set of discrete complex amplitudes multiplying identical local wavefunctions.
Two constants of motion characterize the DNLSE – the system’s energy (Hamiltonian) and the system’s density (number of particles). Site-averaged values of these two quantities place the system as a point on a site-averaged energy-density phase diagram. It turns out that systems on a well-defined area of the phase diagram thermalize. This thermalization property allows the application of statistical mechanics methods to predict equilibrium thermodynamic parameters of systems on the “thermalization zone” [1]. For example – their equilibrium temperatures.
A first paper showing a map of temperatures of systems on the high nonlinearity part of the thermalization zone was already published [2]. We calculated these temperatures through suitably defined temperature-entropy relations.
Since then we have extended our temperature analysis of DNLSE-governed systems to the entire thermalization zone, including the low nonlinearity portion. This was done by adopting a mathematically far more challenging yet rigorous approach – derivations through a suitably defined partition function [1]. Equilibrium distributions of site-densities are numerically calculated through the properties of a transfer integral operator associated with the partition function. The partition-function approach then allows the association of an equilibrium temperature to an equilibrium probability distribution of site densities for thermalized systems at high as well as at low nonlinearities.
Indeed we find somewhat unexpected thermodynamic behavior of low nonlinearity systems. For example - very pronounced rise of temperatures everywhere on the thermalization zone, breaking the variance-temperature relations. Not including the zero-temperature ground state (equal amplitudes alternating signs) that stays at zero temperature at all nonlinearities.
[1] Rasmussen, K. Ø., T. Cretegny, P. G. Kevrekidis, and Niels Grønbech-Jensen, "Statistical mechanics of a discrete nonlinear system." Physical review letters 84, no. 17 (2000): 3740.
[2] Levy Uri, and Yaron Silberberg, "Equilibrium temperatures of discrete nonlinear systems." Physical Review B 98, no. 6 (2018): 060303.
Two constants of motion characterize the DNLSE – the system’s energy (Hamiltonian) and the system’s density (number of particles). Site-averaged values of these two quantities place the system as a point on a site-averaged energy-density phase diagram. It turns out that systems on a well-defined area of the phase diagram thermalize. This thermalization property allows the application of statistical mechanics methods to predict equilibrium thermodynamic parameters of systems on the “thermalization zone” [1]. For example – their equilibrium temperatures.
A first paper showing a map of temperatures of systems on the high nonlinearity part of the thermalization zone was already published [2]. We calculated these temperatures through suitably defined temperature-entropy relations.
Since then we have extended our temperature analysis of DNLSE-governed systems to the entire thermalization zone, including the low nonlinearity portion. This was done by adopting a mathematically far more challenging yet rigorous approach – derivations through a suitably defined partition function [1]. Equilibrium distributions of site-densities are numerically calculated through the properties of a transfer integral operator associated with the partition function. The partition-function approach then allows the association of an equilibrium temperature to an equilibrium probability distribution of site densities for thermalized systems at high as well as at low nonlinearities.
Indeed we find somewhat unexpected thermodynamic behavior of low nonlinearity systems. For example - very pronounced rise of temperatures everywhere on the thermalization zone, breaking the variance-temperature relations. Not including the zero-temperature ground state (equal amplitudes alternating signs) that stays at zero temperature at all nonlinearities.
[1] Rasmussen, K. Ø., T. Cretegny, P. G. Kevrekidis, and Niels Grønbech-Jensen, "Statistical mechanics of a discrete nonlinear system." Physical review letters 84, no. 17 (2000): 3740.
[2] Levy Uri, and Yaron Silberberg, "Equilibrium temperatures of discrete nonlinear systems." Physical Review B 98, no. 6 (2018): 060303.
| Original language | English |
|---|---|
| State | Published - Jul 2019 |
| Event | The 27th International Conference on Statistical Physics of the International Union of Pure and Applied Physics (IUPAP) - Auditorios UCA Puerto Madero, Buenos Aires, Argentina Duration: 8 Jul 2019 → 12 Jul 2019 |
Conference
| Conference | The 27th International Conference on Statistical Physics of the International Union of Pure and Applied Physics (IUPAP) |
|---|---|
| Abbreviated title | StatPhys 27 |
| Country/Territory | Argentina |
| City | Buenos Aires |
| Period | 8/07/19 → 12/07/19 |