UDC 536.758

PHYSICS

V. V. TOLMACHEV

CORRECTIONS \(\Delta F\), \(\Delta n\), AND \(\Delta\mu\) IN THE METHOD OF ASYMPTOTIC EXPANSIONS IN THE MODIFIED FORMULATION OF THE PROBLEM OF A NONIDEAL BOSE–EINSTEIN SYSTEM

(Presented by Academician N. N. Bogolyubov, 10 XI 1966)

In \((^{1-3})\) a method was developed for finding asymptotic expansions in the small interaction in the problem of a nonideal Bose–Einstein system of many particles. It is based on the problem of the asymptotic solution of a certain system of nonlinear integral equations for \(A(p)\), \(B(p)\), and \(\varepsilon(p)\). In the present note an investigation is given of corrections to the results obtained with the aid of the indicated equations.

The nonideal Bose–Einstein system under study, in the modified formulation, is characterized by a Hamiltonian containing two independent parameters \(N_0\), \(\mu\). For this Hamiltonian one must calculate the thermodynamic potential and then set up two equations

\[ \frac{\partial}{\partial N_0} F(N_0,\mu)=0,\qquad \frac{\partial}{\partial \mu} F(N_0,\mu)=-N \tag{1} \]

in order to eliminate from the problem the parameters \(N_0,\mu\). The free energy is calculated from the relation

\[ \Psi = F(N_0,\mu)+\mu N, \tag{2} \]

where on the right the values obtained from (1) must be substituted for \(N_0,\mu\).

The above-mentioned system of nonlinear integral equations arises as a result of carrying out a partial summation of diagrams constructed from pair vertices and proper first-order energy parts. In this case the full \(F(N_0,\mu)\) can be represented in the form

\[ F(N_0,\mu)=F_0(N_0,\mu)+\Delta F(N_0,\mu), \tag{3} \]

where \(F_0\) corresponds to the diagrams included in the partial summation, and \(\Delta F\) corresponds to the remaining diagrams \((^2)\).

Let us give the explicit form of the system of nonlinear integral equations with respect to \(A(p)\), \(B(p)\), \(\varepsilon(p)\), \(\mu\), and \(n_0\):

\[ A(p)=E(p)+n_0 v(p)+\frac{1}{4\pi^2\hbar^3} \int_0^{+\infty} p'^2 dp'\,\bigl(v(p,p')- \]

\[ -\,v(p')\bigr)\left( \frac{A(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2} -1\right)- \]

\[ -\frac{1}{4\pi^2\hbar^3} \int_0^{+\infty} p'^2 dp'\,v(p') \frac{B(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2} -\Delta\mu, \tag{4} \]

\[ B(p)=n_0 v(p)-\frac{1}{4\pi^2\hbar^3} \int_0^{+\infty} p'^2 dp'\,v(p,p') \frac{B(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2}, \tag{5} \]

\[ \varepsilon(p)=\sqrt{A^2(p)-B^2(p)}. \tag{6} \]

\[ \mu = n_0 v(0) + \frac{1}{4\pi^2 \hbar^3}\int_0^{+\infty} p'^2 dp'\,(v(p')+v(0))\left(\frac{A(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2}-1\right) + \]
\[ + \frac{1}{4\pi^2 \hbar^3}\int_0^{+\infty} p'^2 dp'\,\frac{B(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2}+\Delta\mu, \tag{7} \]

\[ n=n_0+\frac{1}{4\pi^2\hbar^3}\int_0^{+\infty} p'^2 dp'\left(\frac{A(p')}{\varepsilon(p')}\operatorname{cth}\frac{\beta\varepsilon(p')}{2}-1\right)+\Delta n, \tag{8} \]

where

\[ \Delta\mu=\Delta\mu(N_0,\mu; A(p),B(p))=\frac{\partial}{\partial N_0}\Delta F(N_0,\mu,A(p),B(p)), \tag{9} \]

\[ \Delta n=\Delta n(N_0,\mu; A(p),B(p))=-\frac{\partial}{\partial \mu}\Delta F(N_0,\mu,A(p),B(p)), \tag{10} \]

where, in turn, \(\Delta F(N_0,\mu; A(p),B(p))\) is a functional of \(A(p), B(p)\), composed of the contributions of the corresponding diagrams of perturbation theory; when taking the derivatives in (9), (10), it is not necessary to affect the functional dependence on \(A(p), B(p)\).

By fixing \(n_0\) instead of \(n\), we somewhat simplify the problem: instead of solving the system (4)—(10), we need solve only the system (4)—(7), (9). True, afterward it is still necessary to find \(n_0\) as a function of \(n\), proceeding from (8), (10), but this is an independent problem.

Equations (4)—(8) differ from (1)—(5) of \({}^{(2)}\) only in that now the corresponding terms have been written explicitly instead of ellipses. In \({}^{(2,3)}\), \(\Delta\mu\) was discarded in (4), and then the resulting system of equations (4)—(6) was solved asymptotically. In (7), (8) the correction terms \(\Delta\mu\), \(\Delta n\) were also discarded, and from the formulas obtained \(\mu\) and \(n\) were calculated. The discarding of \(\Delta\mu, \Delta n\) was justified by the fact that, in comparison with the other terms on the right-hand side of (4), (7), and (8), they are asymptotically less significant.

However, in \({}^{(2,3)}\) attention was not paid to the following circumstance. After, with the aid of (4)—(7) with \(\Delta\mu\) discarded, the terms of the asymptotic expansions for \(A(p,n_0)\), \(B(p,n_0)\), and \(\mu(n_0)\) have been obtained, they can be used to calculate \(\Delta\mu\), in order to verify the smallness of this quantity. This was not done in \({}^{(2,3)}\); instead, the quantity

\[ \Delta F(N_0)=\Delta F(N_0,\mu(n_0); A(p,n_0),B(p,n_0)), \tag{11} \]

or rather its leading asymptotics in the limit of weak interaction, was simply computed, and it was assumed that \(\mu=\dfrac{d}{dN_0}\Delta F/N_0\). However, in order to obtain \(\Delta\mu(N_0)\), knowledge of \(\Delta F(N_0)\) alone is insufficient; one must also know one more quantity,

\[ \Delta n(N_0)=\Delta n(N_0,\mu(n_0); A(p,n_0),B(p,n_0)), \tag{12} \]

or rather its leading asymptotics for weak interaction. Indeed,

\[ \Delta\mu(N_0)=\frac{d}{dN_0}\Delta F(N_0)+\Delta N_0\,\frac{d\mu(n_0)}{dn_0}, \tag{13} \]

so that \(\Delta\mu(N_0)\) is not simply equal to the derivative with respect to \(N_0\) of \(\Delta F(N_0)\).

Let us note that, for the quantities \(\Delta F(N_0)\) and \(\Delta n(N_0)\) defined by (11), (12), many perturbation-theory diagrams possible for the more general quantities \(\Delta F(N_0,\mu; A(p),B(p))\) and \(\Delta n(N_0,\mu; A(p),B(p))\) mutually compensate: there is no need to consider diagrams containing pair vertices or proper first-order self-energy parts.

The leading asymptotic term \(\Delta F(N_0)\) comes from the second-order diagrams schematically shown in Fig. 1. The calculation shows that

\[ \frac{\Delta F(N_0)}{V} = -\frac{1}{3\pi^2\hbar^3}m^{3/2}n_0^{5/2}v^{3/2}(0) \frac{1}{4\pi^2\hbar^3} \int_0^{+\infty} p^2\,dp\,\frac{v^2(p)}{E(p)} + \]

\[ + \frac{1}{16\pi^2\hbar^6}m^3 n_0^3 v^4(0) \left(\frac{4}{3}-\frac{\sqrt{3}}{\pi}\right) \ln v(0) +\text{terms }v^4\text{ and higher}, \tag{14} \]

in the case of zero temperature \(\theta=0\), and

\[ \frac{\Delta F(N_0)}{V} = \frac{n_0}{2\pi^2\hbar^6}m^3 v^2(0)\theta^2 \ln\frac{n_0 v(0)}{\theta} +\text{terms }v^2\text{ and higher} \tag{15} \]

in the general case \(\theta\ne 0\)*. The calculation of the leading asymptotic term \(\Delta n(N_0)\) is more complicated. It is now necessary to take into account an entire subsequence of diagrams, the first of which are shown in Fig. 2.

Fig. 1

Fig. 2

In the case \(\theta\ne 0\) we obtain

\[ \Delta n = -\frac{1}{(2\pi^2\hbar^3)^{3/2}} m^{3/4}n_0^{1/4}v^{3/4}(0)\theta^{3/2} +\text{terms higher than }v^{3/4}, \tag{16} \]

where exactly one half of the result comes from the first two second-order diagrams. Thus we find that the correction \(\Delta n\) exactly compensates the correction term \(v^{3/4}\) in formula (9) of (2). The remaining formulas in (2) require no corrections.

Moscow State University
named after M. V. Lomonosov

Received
19 IX 1966

CITED LITERATURE

\(^{1}\) V. V. Tolmachev, DAN, 134, 1324 (1960).
\(^{2}\) V. V. Tolmachev, DAN, 135, 41 (1960).
\(^{3}\) V. V. Tolmachev, DAN, 135, 825 (1960).

* In (2), the coefficient in formula (15) is incorrect. In addition, in (2) there is an inaccuracy in Fig. 2.