Systems at Finite Temperature

Garry Bowen, M. Gulácsi

Finite-temperature properties of a non-Fermi-liauid system is one of the most challenging problems in the current understanding of strongly correlated electron systems. The paradigmatic arena for studying non-Fermi liauids is in one dimension, where the concept of a Luttinger liauid has arisen. The existence of a critical point at zero temperature in one-dimensional systems, and the fact that experiments are all undertaken at finite temperatures, implies a need for these one-dimensional systems to be examined at finite temperatures. Accordingly, we extended the well-known bosonization method of one-dimensional electron systems to finite temperatures. We have used this new bosonization method to calculate finite-temperature asymptotic correlation functions for linear fermions, the Tomonaga-Luttinger model, and the Hubbard model.

Yuan Du-Qi

At zero-temperature and finite-temperature, the thermodynamic properties of finite unitary Fermi gas in a three-dimensional harmonic trap are investigated by using fractional exclusion statistics, and the results are compared with those of the system which satisfies the thermodynamic limit. At zero-temperature, Fermi energy and average energy of per particle increase with the increase of the number of particles for finite unitary Fermi gas, and their limits are the corresponding parameters of the system which satisfy thermodynamic limits. Fermi energy and average energy of per particle each have a maximum value changing with the boundary of the potential well. For the finite-temperature trapped unitary Fermi system, when the number of particles is certain the average energy of per particle, average entropy of per particle, average heat capacity of per particle each have a characteristic temperature, respectively, when the temperature is equal to the characteristic temperature of the physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the temperature is lower (or higher) than the characteristic temperature of parameter, the physical parameter of the finite system will be greater (or less) than the corresponding parameter of the thermodynamic limit system. The characteristic temperature has particle number effect and boundary effect. When the temperature is determined, the average energy of per particle, average entropy of per particle and average heat capacity of per particle each have a characteristic number of particles, respectively, when the number of particles is equal to the characteristic number of particles for physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the number of particles is less (or more) than the characteristic number of particles for corresponding parameter, the corresponding parameter of the finite system will be less (or larger) than the thermodynamic limit of system.

Yoshimasa Nakano

Shaoyong Zhang, Zhaoyu Fei, Xiaoguang Wang

Abstract Temperature of a finite-sized system fluctuates due to the thermal fluctuations. However, a systematic mathematical framework for measuring or estimating the temperature is still underdeveloped. Here, we incorporate the estimation theory in statistical inference to estimate the temperature of a finite-sized system and propose optimal estimation based on the uniform minimum variance unbiased estimation. Treating the finite-sized system as a thermometer measuring the temperature of a heat reservoir, we demonstrate that different optimal estimation of parameters yield different formulas of entropy, e.g., optimal estimation of inverse temperature (or temperature) aligns with the Boltzmann entropy (or Gibbs entropy). The optimal estimation leads to a achievable energy-temperature uncertainty relation and exhibits sample-size dependence, coinciding with their counterparts in nanothermodynamics. The achievable bound and the non-Gaussian distribution of temperature enable experimental testing in finite-sized systems.