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Problems on non-equilibrium statistical physics

Moochan B. Kim

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Four problems in non-equilibrium statistical physics are investigated: 1. The\nthermodynamics of single-photon gas; 2. Energy of the ground state in Multi-electron\natoms; 3. Energy state of the H2 molecule; and 4. The Condensation behavior in N\nweakly interacting Boson gas.\nIn the single-photon heat engine, we have derived the equation of state similar\nto that in classical ideal gas and applied it to construct the Carnot cycle with a single\nphoton, and showed the Carnot efficiency in this single-photon heat engine.\nThe energies of the ground state of multi-electron atoms are calculated using the\nmodi ed Bohr model with a shell structure of the bound electrons. The di erential\nSchrodinger equation is simpli ed into the minimization problem of a simple energy\nfunctional, similar to the problem in dimensional scaling in the H-atom. For the\nC-atom, we got the ground state energy -37:82 eV with a relative error less than 6\n%.\nThe simplest molecular ion, H+\n2 , has been investigated by the quasi-classical\nmethod and two-center molecular orbit. Using the two-center molecular orbit derived\nfrom the exact treatment of the H+\n2 molecular ion problem, we can reduce the number\nof terms in wavefunction to get the binding energy of the H2 molecule, without using\nthe conventional wavefunction with over-thousand terms. We get the binding energy\nfor the H2 with Hylleraas correlation factor 1 + kr12 as 4:7eV, which is comparable\nto the experimental value of 4:74 eV.\nCondensation in the ground state of a weakly interacting Bose gas in equilibrium\nis investigated using a partial partition function in canonical ensemble. The recursive\nrelation for the partition function developed for an ideal gas has been modi ed to\nbe applicable in the interacting case, and the statistics of the occupation number in\ncondensate states was examined. The well-known behavior of the Bose-Einstein Condensate\nfor a weakly interacting Bose Gas are shown: Depletion of the condensate\nstate, even at zero temperature, and a maximum \nuctuation near transition temperature.\nFurthermore, the use of the partition function in canonical ensemble leads to\nthe smooth cross-over between low temperatures and higher temperatures, which has\nenlarged the applicable range of the Bogoliubov transformation. During the calculation,\nwe also developed the formula to calculate the correlations among the excited\nstates.

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Four problems in non-equilibrium statistical physics are investigated: 1. The\nthermodynamics of single-photon gas; 2. Energy of the ground state in Multi-electron\natoms; 3. Energy state of the H2 molecule; and 4. The Condensation behavior in N\nweakly interacting Boson gas.\nIn the single-photon heat engine, we have derived the equation of state similar\nto that in classical ideal gas and applied it to construct the Carnot cycle with a single\nphoton, and showed the Carnot efficiency in this single-photon heat engine.\nThe energies of the ground state of multi-electron atoms are calculated using the\nmodi ed Bohr model with a shell structure of the bound electrons. The di erential\nSchrodinger equation is simpli ed into the minimization problem of a simple energy\nfunctional, similar to the problem in dimensional scaling in the H-atom. For the\nC-atom, we got the ground state energy -37:82 eV with a relative error less than 6\n%.\nThe simplest molecular ion, H+\n2 , has been investigated by the quasi-classical\nmethod and two-center molecular orbit. Using the two-center molecular orbit derived\nfrom the exact treatment of the H+\n2 molecular ion problem, we can reduce the number\nof terms in wavefunction to get the binding energy of the H2 molecule, without using\nthe conventional wavefunction with over-thousand terms. We get the binding energy\nfor the H2 with Hylleraas correlation factor 1 + kr12 as 4:7eV, which is comparable\nto the experimental value of 4:74 eV.\nCondensation in the ground state of a weakly interacting Bose gas in equilibrium\nis investigated using a partial partition function in canonical ensemble. The recursive\nrelation for the partition function developed for an ideal gas has been modi ed to\nbe applicable in the interacting case, and the statistics of the occupation number in\ncondensate states was examined. The well-known behavior of the Bose-Einstein Condensate\nfor a weakly interacting Bose Gas are shown: Depletion of the condensate\nstate, even at zero temperature, and a maximum \nuctuation near transition temperature.\nFurthermore, the use of the partition function in canonical ensemble leads to\nthe smooth cross-over between low temperatures and higher temperatures, which has\nenlarged the applicable range of the Bogoliubov transformation. During the calculation,\nwe also developed the formula to calculate the correlations among the excited\nstates.
A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number NNN of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N=200N=200N=200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.