Statistical Description of Fermi System over a Surface in a Uniform External Field

The gist The statistical description of Fermi system over a surface in a uniform external field is proposed and its thermodynamic characteristics are found, in particular heat capacities under various conditions.

System Setup and Energy Spectrum

The analysis considers a system of an arbitrary number of Fermi particles in a constant uniform external field located over a flat surface of finite size, where the uniform field is directed along the z axis as U(z) = F z (1). The solution to the Schrödinger equation leads to energy levels determined by the condition for the wave function on the surface Ai(−εn) = 0 (13). The full energy spectrum of a fermion is given by E(i,n) = 2/2m 2π/L squared γ squared i + 2/ml squared εn (17). The discrete energy levels εn are found from the condition cos 2/3 ε(3/2 n − 1/4) = 0 (16).

Statistical Description and Thermodynamic Potentials

The statistical description proceeds from the formula for entropy S = P ν Sν, where Sν = ln Γ(ri + 1) − ln Γ(rifν + 1) − ln Γ ri(1 − fν) + 1 (18). The total energy of the whole system is E = X ν Eν fνri (19), and the average number of particles at level ν is found from the condition ∂/∂fν S − αN − βE = 0, yielding εn θ(fν, ri) ≡ ψ ri(1 − fν) + 1− ψ rifν + 1 = (Eν − µ)/T (22). The thermodynamic potential is defined by the usual formula omega = E − T S − µN (23), and its differential takes the form domega = X ν rifν dEν − SdT − N dµ (23).

Surface Tension and Induction

The differential of the thermodynamic potential can be represented in the form domega = -SdT − N dµ + σdA + DdF (23). The quantity σ = ∂omega/∂A T,µ,F has the meaning of the surface tension, and D = ∂omega/∂F T,µ,A can naturally be called “induction” (27). The differentials for entropy (30), surface tension (-dσ 1/πT A 2) (31), and induction dD are derived using various notations involving quantities like χ1, χ2, etc.

Heat Capacities in the Continuum Approximation

In the continuum approximation, where the area A is large and L → ∞, the number of states in a large system is related to the area of a circle S(γ) = πγ squared (4). The total number of particles at level (n) with use of Fermi-Dirac distribution (60) is Nn = 2/L Λ squared Z ∞ 0 dy e(y−tn + 1) = 2/L Λ squared Φ1(tn) (62). In this approximation, the heat capacities are given by formulas derived from the continuum description of thermodynamic quantities (67). For constant number of particles dN = 0, the heat capacity under arbitrary conditions is TdS/dT = πΛ squared η2T dA/dT − 2l cubed η4dF/dT + η1 (41), and for fixed surface tension dσ = 0 and field dF = 0, CN,F,σ = πΛ squared η2 / (A Λ 2) - πΛ squared η3 / A − 1 + πΛ squared η6 (44).

Results in the Continuum Limit

The general relations simplify to expressions for heat capacities that depend on three independent parameters η3, η5, and η6 (50). The heat capacities under various conditions are given by CN,A,F = π/2 l 4/Lη3 + 1/2 l 2Ll squared F(4π) squared η6 + 1/2 l 2Ll squared F η5 (51), CN,F,σ = πΛ A 1/π T A L cubed / (A Λ 2) - πΛ A 1/π T A L cubed / (A Λ 2) + 1/2 l 2Ll squared F η5 (52), and CN,F,D = 1/4η5 l 2Ll squared F η i − η3η6 (53). The heat capacities per one particle are then expressed through standard functions Φs(t) (79), where Ψ1(t) ≡ Φ7/2(t)Φ5/2(t)−5/7Φ5/2(t)Φ3/2(t), Ψ2(t) ≡ Φ25/2(t).

Field Dependence and Limits

The dependences of heat capacities on the field intensity E and temperature T are obtained in Fig. 6 (6). At low temperatures, the heat capacities are proportional to temperature, cF,A = cF,σ = cF,D ∼ 2π/23 nAl squared T/TB (96). At high temperatures, the heat capacity at constant induction increases as cF,D ∼ T 5/2 (83), while the other two heat capacities tend to constant values cF,A = 5/2 and cF,σ = 7/2 (84). The fermion density distribution over the surface is found in the continuum limit when formula (62) is valid, showing a maximum near a small distance from the surface at zero temperature (88).

Application to External Fields

The general relations can be applied to gravitational and electric fields, with minimum temperatures determined by characteristic lengths l = ΛB ≡ 2π 2/mTB 1/2 (91). For the electric field, the dimensionless surface density nAl squared slowly increases with the field intensity nAl squared = nA∗l 2∗ E/E∗ 1/3 (94). The dependences of heat capacities in the electric field on temperature and field intensity are obtained in Fig. 5 (6). In a strong field, all particles in a strong field accumulate at the lower level (Fig. 6).

Conclusion

A statistical approach to the description of the thermodynamic properties of the Fermi particle system over a flat surface in a uniform external field is proposed and its thermodynamic characteristics are found, in particular heat capacities under various conditions. The cases of gravitational and electric fields are considered (VI). The continuum approximation is considered, in which the surface area is assumed to be large, so that the motion in the plane is characterized by a two-dimensional wave vector (IV). Near zero temperature, the heat capacities are proportional to temperature (VI). At high temperatures, the heat capacity at constant induction increases as cF,D ∼ T 5/2 (VI). The fermion density distribution over the surface is found (VI). The cases of gravitational and electric fields are considered in this paper.

--- Page 1 ---

The statistical description of Fermi system over a surface in a uniform external field is proposed and its thermodynamic characteristics are found, in particular heat capacities under various conditions. The number of particles per unit area is assumed to be arbitrary, in particular, small. General formulas are obtained for entropy, energy, thermodynamic potential, heat capacities under various conditions and the distribution of the particle number density over the surface. In the continuum limit of a large surface area, the temperature dependences of heat capacities and density distribution are calculated. The cases of gravitational and electric fields are considered. General formulas for the heat capacities at fixed surface area, surface tension and induction are obtained. A continuum approximation is considered, in which the surface area is assumed to be large, so that the motion in the plane is characterized by a two-dimensional wave vector. Near zero temperature, the heat capacities are proportional to temperature. At high temperatures, the heat capacity at constant induction increases as cF,D ∼ T 5/2, and the other two heat capacities tend to constant values cF,A = 5/2 and cF,σ = 7/2. The fermion density distribution over the surface is found. The cases of gravitational and electric fields are considered in this paper.>

How it works

The statistical description proceeds from the formula for entropy S = P ν Sν, where Sν = ln Γ(ri + 1) − ln Γ(rifν + 1) − ln Γ ri(1 − fν) + 1 (18).

1. Quantum Dot/Nanostructure Thermodynamics: The improved AI system can accurately describe the thermodynamic properties of such objects, taking into account their interaction with the external environment and in external fields, specifically by calculating entropy, energy, thermodynamic potential, heat capacities under various conditions and the distribution of the particle number density over the surface.

2. Continuum Limit Calculations: The system can perform calculations in the continuum limit of a large surface area, enabling it to determine the temperature dependences of heat capacities and density distribution for systems with arbitrary particle numbers.

3. Gravitational and Electric Field Modeling: The AI can apply general formulas derived for the continuum limit to calculate thermodynamic properties under specific external fields, such as gravitational and electric fields.

4. Field-Dependent Heat Capacity Prediction: Given a fixed temperature and field intensity, the system can predict the heat capacities based on relations like CN,F,A = π2/l4Lη3 + l2F/4π(4π)2η6 + 1/2l2Ll2F η5 (Equation 51).

5. High-Temperature Asymptotic Behavior: The AI can predict the high-temperature behavior of heat capacities, such as cF,A = 5/2 − 15/32√2α T 4 / TB 4, where α is related to surface density.

6. Density Distribution Mapping: The system can determine the fermion density distribution over the surface by calculating n(z) = 1/L2 Xν ψ squared n(z)fνri (Equation 54), allowing it to map particle concentration across the surface for various temperatures and field intensities.

A statistical approach to the description of the thermodynamic properties of the Fermi particle system occupying a half-space over a plane of finite size in a uniform external field is proposed. The number of particles per unit area is assumed to be arbitrary, in particular, small. General formulas are obtained for entropy, energy, thermodynamic potential, heat capacities under various conditions and the distribution of the particle number density over the surface. In the continuum limit of a large surface area, the temperature dependences of heat capacities and density distribution are calculated. The cases of gravitational and electric fields are considered.