Operators of Time Evolution S(t, t0) and Sα(t, t0)

Maurizio Fagotti

We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites L^d L d . We compute the distribution of the eigenvalues of the time averaged state over a time window [t_0,t_0+t] [ t 0 , t 0 + t ] in the limit of large L. This allows us to infer the size of a subspace that captures time evolution in [t_0,t_0+t] [ t 0 , t 0 + t ] with an accuracy 1-\epsilon 1 − ϵ . We estimate the size to be \frac{\sqrt{2{\mathfrak e}_2}}{\pi}erf^{-1}(1-\epsilon) L^{\frac{d}{2}}t 2 𝔢 2 π e r f − 1 ( 1 − ϵ ) L d 2 t , where {\mathfrak e}_2 𝔢 2 is the energy variance per site, and erf^{-1} e r f − 1 is the inverse error function.

Ilya Prigogine, Evgueni Karpov, Tomio Petrosky +1 more

Gebhard Grübl, C. Leubner

Márcio Freire

In this work we will establish the definitions of the Green’s functions in statistical mechanics and their basic properties. These functions depend on double-time and temperature. It can be observed by mean of theier definitions, where we can find the medium values of the operators product, this average is done over the grand canonical ensemble, for instance, (FAB(t − t0) ≡ hAˆ(t)Bˆ(t0)i ou FBA(t − t0) ≡ hBˆ(t0)Aˆ(t)i), with β = 1/kBT, and T is the absolute temperature. The operators involved in theses functions satisfy the Heisenberg equation of motion, which permit us describe the evolution equations for Green’s functions. By mean of spectral representation of the time correlation functions(FAB(t−t0) ≡ hAˆ(t)Bˆ(t0 )i or FBA(t−t0) ≡ hBˆ(t0)Aˆ(t)i), that is done by the introduction of a Fourier transform to change the system from time-space to frequency-space, we can obtain the spectral representations for the retarded Gr, advanced Gr e causal Gc green’s functions. Lastly, we will make the use of retarded green’s function to describe the electrical conductivity of a system of electrons under a external time-dependent electric field, in others words, we will write the electrical conductivity tensor in terms of Gr and, at last, we will calculate the electrical conductivity of a system of electrons and phonons.

S.R. DE GROOT, G.J. Hooyman, Cornelis A. Ten Seldam

Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.

Deng-yuan Chen