Abstract
Full Text
PHYSICS
I. B. ALEKSANDROV, Yu. A. KUKHARENKO, A. V. NIUKKANEN
KINETIC EQUATION OF A NONIDEAL FERMI SYSTEM
(Presented by Academician N. N. Bogolyubov, 11 September 1962)
- With the aid of Bogolyubov’s method of the generating functional \({}^{(1)}\), generalized to the case of quantum statistics \({}^{(3)}\), for the \(s\)-particle density operators \(F_s = \operatorname{Sp}_{(s+1,\ldots,N)} \rho(1,\ldots,N)\) (\(\rho\) is the density matrix) a chain of coupled equations is obtained:
\[ \frac{\partial F_s}{\partial t} = \left[ \sum_{i=1}^{N} T_i + \sum_{1\leq j<r\leq N} \Phi(j,r);\, F_s \right] + \frac{1}{v}\operatorname{Sp}_{(s+1)} \left[ \sum_{1\leq k\leq s} \Phi(k,s+1);\, F_{s+1} \right]. \tag{1} \]
The matrix of the potential \(\Phi\) of the pair interaction of particles is not assumed to be diagonal. In order to satisfy the law of conservation of momentum, we require that the matrix elements \(\Phi(r_1 r_2, r'_1 r'_2)\) be invariant with respect to spatial translations. In this case
\[ \Phi(r_1,r_2; r'_1,r'_2)= \]
\[ = \int \widetilde{\Phi}(p_1,p_2;p'_1p'_2) \exp\{ i(p_1r_1+p_2r_2-p'_1r'_1-p'_2r'_2)\} \,dp_1dp_2dp'_1dp'_2, \]
\[ \widetilde{\Phi}(p_1,p_2;p'_1,p'_2) = \Phi(p_1,p_2;p'_1,p'_2)\, \delta(p_1+p_2-p'_1-p'_2). \tag{2} \]
By virtue of the Hermiticity of the Hamiltonian and the symmetry properties of the system,
\[ \Phi(p_1,p_2;p'_1,p'_2) = \Phi^{*}(p'_1,p'_2;p_1,p_2), \qquad \Phi(p_1,p_2;p'_1,p'_2) = \Phi(p_2,p_1;p'_2,p'_1). \tag{3} \]
We shall consider the system of equations (1) under the assumption that the potential energy of interaction of a pair of particles is small in comparison with their mean kinetic energy. In this case a small parameter of perturbation theory \({}^{(2)}\) is introduced: \(\Phi \to \varepsilon \Phi,\ \varepsilon \ll 1\). According to \({}^{(2)}\), we introduce the operator \(g\)
\[ F_2(t;p_1,p_2;p'_1,p'_2) = \gamma'_2 F_1(p_1,p'_1;t) F_1(p_2,p'_2;t) + \varepsilon g(p_1,p_2;p'_1,p'_2;t), \]
\[ F_3(t,p_1,p_2,p_3;p'_1,p'_2,p'_3) - \gamma'_3 F_1(p_1,p'_1,t)F_1(p_2,p'_2,t)F_1(p_3,p'_3,t) + \]
\[ + \varepsilon \gamma'_{13} g(p_1,p_2;p'_1,p'_2;t) F_1(p_3,p'_3), \tag{4} \]
where \(\gamma_s=\sum_{(P)}(-1)^P\) denotes the antisymmetrized sum over all \(s!\) permutations of primed indices. This method makes it possible to obtain an equation for \(F_1\) with no divergent terms in the second order. To obtain a closed equation for \(F_1\) on the basis of approximation (4), it is necessary to find the functional dependence of the correlation operator \(g\) on \(F_1\). Substituting (4) into equations (1) for \(F_1\) and \(F_2\), written in the momentum representation, one can show that \(g\) satisfies the equation
\[ i\hbar \frac{\partial g(p_1,p_2;p'_1,p'_2,t)}{\partial t} - (T_{p_1}+T_{p_2}-T_{p'_1}-T_{p'_2}) g(p_1p_2,p'_1p'_2,t) + \]
\[ + A(p_1,p_2;p'_1,p'_2;F_1(t)), \tag{5} \]
\[ T_{p_i}=\frac{p_i^2}{2m}. \]
\[ \begin{aligned} A(p_1,p_2;p'_1,p'_2;F_1(t))={}&\int dp''_1\,dp''_2\Bigg\{L_1(p_1,p_2;p''_1,p''_2;p'_1p'_2)+\\ &+\frac{1}{v}\int dp_3\,dp''_3\,L_2(p_1,p_2,p_3;p''_1,p''_2,p''_3;p'_1,p'_2,p_3)\Bigg\},\\ L_1={}&\tilde{\Phi}(p_1,p_2;p''_1,p''_2)\gamma''_2F_1(p''_1,p'_1)F_1(p''_2,p'_2)-\\ &-\tilde{\Phi}(p''_1,p''_2;p'_1,p'_2)\gamma'_2F_1(p_1,p''_1)F_1(p_2,p''_2), \end{aligned} \tag{6} \]
\[ \begin{aligned} L_2={}&\{\tilde{\Phi}(p_1,p_3;p''_1,p''_3)\delta(p_2-p''_2) +\tilde{\Phi}(p_2,p_3;p''_2,p''_3)\delta(p_1-p''_1)\}\times\\ &\times\gamma''_3F_1(p''_1,p'_1)F_1(p''_2,p'_2)F_1(p''_3,p_3)-\\ &-\{\tilde{\Phi}(p''_1,p''_3;p'_1,p_3)\delta(p'_2-p''_2) +\tilde{\Phi}(p''_2,p''_3;p'_2,p_3)\delta(p'_1-p''_1)\}\gamma'_3\times\\ &\times F_1(p_1,p''_1)F_1(p_2,p''_2)F_1(p_3,p''_3). \end{aligned} \]
Let us impose on the solution of equation (5) the boundary condition of weakening of correlations:
\[ g(t)\to 0\quad \text{as}\quad t\to-\infty, \tag{7} \]
used in work \((^4)\). This condition proves to be productive in the present work. The solution of equation (5), taking into account the boundary condition (7), has the form:
\[ g(p_1,p_2;p'_1,p'_2,t)= \tag{8} \]
\[ =\int_{-\infty}^{t}d\tau\,A(p_1,p_2;p'_1,p'_2;\tau) \exp\left\{\frac{i}{\hbar}\bigl(T(p_1)+T(p_2)-T(p'_1)-T(p'_2)\bigr)(t-\tau)\right\}. \]
Taking into account the approximation (4) and substituting the solution found for \(g\) into the equation for \(F_1\), we obtain the kinetic equation for a system of fermions with nonlocal interaction in the general spatially inhomogeneous case:
\[ \begin{aligned} i\hbar\frac{\partial F_1(t,p_1,p'_1)}{\partial t} ={}&(T_{p_1}-T_{p'_1})F_1(t,p_1,p'_1)+\\ &+\frac{\varepsilon}{v}\int dp_2\,dp''_1\,dp''_2\,L_1(p_1,p_2;p''_1,p''_2;p'_1,p_2,F_1(t))+\\ &+\frac{\varepsilon^2}{v}\int dp_2\,dp''_1\,dp'_2 \{\tilde{\Phi}(p_1,p_2;p''_1,p''_2)g(p''_1,p''_2;p'_1,p_2)-\\ &\qquad\qquad\qquad\qquad -\tilde{\Phi}(p''_1,p''_2;p'_1,p_2)g(p_1,p_2;p''_1,p''_2)\}, \end{aligned} \tag{9} \]
where \(g\) is given by formula (8).
- Consider small deviations from the equilibrium spatially homogeneous distribution of the system of Fermi particles:
\[ F_1(t,p_1,p'_1)=F_0(p_1,p'_1)+\delta F_1(t,p_1,p'_1), \tag{10} \]
where \(F_0(p_1,p'_1)=vn(p_1)\delta(p_1-p'_1)\) is a certain equilibrium function of the spatially homogeneous distribution, and \(n(p_1)\) are the occupation numbers of states with given momenta and spins.
Substituting (10) into (9) and neglecting terms containing, as a factor, a quantity of second order of smallness \((\delta F_1)^2\), we obtain a variational linearized equation for the small deviation \(\delta F_1\). Represent \(\delta F_1\) in the form
\[ \delta F_1(tp_1p'_1)=f_q(p_1,t)\delta(p_1-p'_1-q). \tag{11} \]
It can be shown that \(f_q(p_1,t)\) satisfies the integral equation:
\[
i\hbar \frac{\partial f_q(p_1 t)}{\partial t}
=
(T_{p_1}-T_{p_1-q})f_q(p_1,t)+
\]
\[
+\varepsilon \int dp_2 \{R(p_1,p_2)[B(p_1p_2,p_1p_2)+B(p_1-q,p_2,p_1-q,p_2)]+
\]
\[
+B(p_1p_2,p_1-q,p_2+q)[R(p_2,p_1)+R(p_2,p_1-q)]\}+
\]
\[
+\varepsilon^2\int dp'_1 \int_{-\infty}^{t}d\tau \int dp_2\,dp''_1\,dp''_2
\{\widetilde{\Phi}(p_1p_2,p''_1p''_2)\times
\tag{12}
\]
\[
\times \exp\left(\frac{i}{\hbar}(T_{p''_1}+T_{p''_2}-T_{p'_1}-T_{p_2})'(t-\tau)\right)
\delta A(p''_1p''_2,p'_1,p_2,\tau)-
\]
\[
-\widetilde{\Phi}(p''_1p''_2,p'_1p_2)
\exp\left(\frac{i}{\hbar}(T_{p_1}+T_{p_2}-T_{p''_1}-T_{p''_2})(t-\tau)\right)
\cdot \delta A(p_1p_2,p''_1p''_2,\tau),
\]
where the following notation has been introduced:
\[ R(p_1,p_2)=f_q(q_1)n(p_2),\qquad Q(p_1,p_2p_3)=f_q(p_1)n(p_2)n(p_3), \]
\[ B(p_1p_2,p'_1p'_2)=\Phi(p_1p_2,p'_1p'_2)-\Phi(p_1p_2,p'_2p'_1). \]
The varied term \(\delta A(p_1p_2,p'_1p'_2)\) represents the following sum:
\[
\delta A(p_1p_2,p'_1p'_2)
=
A_1(p_1p_2,p'_1p'_2)(p_1+p_2-p'_1-p'_2-q)+
\]
\[
+G_1(p_1,p_2)\,[\delta(p_2-p'_2)\delta(p_1-p'_1-q)
-\delta(p_2-p'_1)\delta(p_1-p'_2-q)]+
\]
\[
+G_2(p_1,p_2)\,[\delta(p_1-p'_1)\delta(p_2-p'_2-q)
-(p_1-p'_2)\delta(p_2-p'_1-q)];
\tag{13}
\]
\[
A_1(p_1p_2,p'_1p'_2)
=
B(p_1p_2,p'_1+q,p'_2)\alpha_1(p_1p_2,p'_1p'_2)f_q(p'_1+q)+
\]
\[
+B(p_1p_2,p'_1,p'_2+q)\alpha_2(p_1p_2,p'_1p'_2)f_q(p'_2+q)-
\]
\[
-B(p_1-q,p_2,p'_1p'_2)\beta_1(p_1p_2,p'_1p'_2)f_q(p_1)
-
B(p_1,p_2-q,p'_1p'_2)\beta_2(p_1p_2,p'_1p'_2)f_q(p_2),
\]
\[
\alpha_1(p_1p_2,p'_1p'_2)
=
\alpha_2(p_2p_1,p'_2p'_1)
=
-\beta_1(p'_1p'_2,p_1p_2)
=
-\beta_2(p'_2p'_1,p_2p_1)
=
\]
\[
=n_{p'_2}+n_{p_1}n_{p_2}-n_{p_2}n_{p'_2}-n_{p_1}n_{p'_2};
\tag{14}
\]
\[
G_1(p_1,p_2)=G_2(p_2,p_1)=
\]
\[
=\frac{1}{v}\int
\Bigl(
Q(p_1,p_2p_3)\{B(p_1p_3,p_1p_3)+B(p_1-q,p_3,p_1-q,p_3)+2B(p_2p_3,p_2p_3)\}
+
\]
\[
+B(p_1p_3,p_1-q,p_3)\{Q(p_3,p_2,p_1-q)+Q(p_3+q,p_2,p_1)\}
\Bigr)\,dp_3.
\tag{15}
\]
In what follows we shall consider the case where the momentum is sufficiently small in the sense that the terms of the kinetic equation proportional to \(\varepsilon^2 q\) are of higher order of smallness than the terms proportional to \(\varepsilon^2\). Since the kinetic equation is derived in the approximation quadratic in \(\varepsilon\), in \(\delta A\) one may put \(q=0\), which corresponds to taking into account the leading term of the expansion of \(\delta A\) in powers of \(q\).
In the approximation under consideration, \(A_1\) takes the form
\[ A_1(p_1,p_2;p'_1,p'_2) = B(p_1,p_2;p'_1,p'_2)\, S(p_1,p_2;p'_1,p'_2;f_q), \tag{16} \]
where
\[
S(p_1,p_2;p'_1,p'_2;f_q)
=
\alpha_1(p_1,p_2;p'_1,p'_2)f_q(p'_1)+
\]
\[
+\alpha_2(p_1,p_2;p'_1,p'_2)f_q(p'_2)
+\beta_1(p_1,p_2;p'_1,p'_2)f_q(p_1)
+\beta_2(p_1,p_2;p'_1,p'_2)f_q(p_2).
\]
If one takes into account that the coefficient satisfies the symmetry property
\[ S(p_1,p_2;p'_1,p'_2) = S(p_1,p_2;p'_2,p'_1) = S(p_2,p_1;p'_1,p'_2) = -S(p'_1,p'_2;p_1,p_2), \]
then the quadratic term of equation (12) can be symmetrized by means of the mutual interchange of variables \(p'_1 \leftrightarrow p'_2\), and in the special case of slow change
change of \(F_1(t)\) in time in comparison with \(F_2(t)\) takes the form of the varied collision term
\[ \varepsilon^2 I_B(f_q) = 2\pi \hbar \varepsilon^2 \int dp_2\,dp_1''\,dp_2'' \left|B(p_1,p_2;p_1'',p_2'')\right|^2 S(p_1'',p_2'';p_1,p_2) \times \]
\[ {}\times \delta(p_1+p_2-p_1''-p_2'') \delta(T_{p_1}+T_{p_2}-T_{p_1''}-T_{p_2''}), \tag{17} \]
where \(\left|B(p_1,p_2;p_1'',p_2'')\right|^2\) is the transition probability between states with different momenta, and
\[ S(p_1'',p_2'';p_1,p_2) = \delta\{n_{p_1''}n_{p_2''}(1-n_{p_1})(1-n_{p_2}) - n_{p_1}n_{p_2}(1-n_{p_1''})(1-n_{p_2''})\}. \]
The functions \(n(p_1-q)\), \(B(p_1-q,p_2,p_2,p_1-q)\), which enter the self-consistent-field term, may, for sufficiently small \(q\), be approximately replaced by the first two terms of their expansions in power series in a neighborhood of \(q=0\). Neglecting terms proportional to \(\varepsilon q^2\), the equation is considerably simplified. In the particular case of a diagonal potential
\[ \widetilde{\Phi}(p_1,p_2;p_1',p_2') = \Phi(p_1-p_2')\delta(p_1+p_2-p_1'-p_2') \]
we obtain, without taking into account the term \(\varepsilon^2 I_B\), the equation with the self-consistent field \({}^{(5,6)}\):
\[ i\hbar\frac{\partial f_q(p_1,t)}{\partial t} = \left\{ \frac{p_1 q}{m} - \varepsilon q \int dp_2\,\frac{\partial n_{p_2}}{\partial p_2}\Phi(p_1-p_2) \right\} f_q(p_1,t) + \varepsilon q \frac{\partial n_{p_1}}{\partial p_1} \int dp_2 f_q(p_2,t) \times \]
\[ {}\times \bigl(\Phi(p_1-p_2)-\Phi(q)\bigr) + \varepsilon^2 I_B(f_q). \tag{18} \]
If, in deriving the kinetic equation, one restricts oneself to terms linear in \(\varepsilon\), then after the replacement \(p_1\to p_1+q/2\) the equation of work \({}^{(6)}\) can be obtained from (12) without the assumption that \(q\) is small.
To find the zeroth approximation it is necessary to take into account the term caused by pair correlation \({}^{(7)}\), which is responsible for establishing the equilibrium distribution. We note that a nonequilibrium spatially homogeneous distribution
\[ F_0(t,p_1,p_1') = v n(p_1,t)\delta(p_1-p_1') \]
may be chosen as the zeroth approximation; as is seen from (12), it satisfies the equation derived in \({}^{(4)}\).
From equation (18) one finds \({}^{(5)}\) the dispersion equation for the zero-sound spectrum, containing, besides the “self-consistent” damping, the damping \(\gamma_{st}\) caused by collisions of particles. For small \(q\), the “self-consistent” damping may be neglected \({}^{(8)}\).
Taking the frequency of the spectrum to be much greater than the collision frequency \(1/\tau\) (\(\tau\) is the mean free time of the particles) and taking into account that \(I_B\sim \varepsilon^2\), we obtain:
\[ \gamma_{st} = \varepsilon^2 \int dp_2\,dp_1''\,dp_2'' \left|B(p_1,p_2;p_1'',p_2'')\right|^2 \alpha_1(p_1'',p_2'';p_1,p_2) \times \]
\[ {}\times \delta(T_{p_1}+T_{p_2}-T_{p_1''}-T_{p_2''}) \delta(p_1+p_2-p_1''-p_2''). \]
The vanishing of this expression at absolute zero temperature does not mean the absence of damping, since, as noted above, the expansion in powers of \(q\) of functions that undergo a discontinuity at the Fermi surface is not mathematically justified at \(T=0\).
The authors express their sincere gratitude to Academician N. N. Bogolyubov for his constant attention to the work and for useful advice.
Moscow State University
named after M. V. Lomonosov
Received
18 VII 1962
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