Abstract Generated abstract
This paper derives kinetic equations for weakly interacting, weakly excited superfluid Bose systems and solves them in the hydrodynamic approximation. Starting from a Bogolyubov quasiparticle Hamiltonian with external sources, the author formulates equations of motion for two-time temperature Green functions, applies controlled decouplings and first-order approximations in the interaction parameter, and obtains a kinetic equation with drift and collision terms. Using a Chapman-Enskog expansion, the zeroth approximation yields hydrodynamic poles corresponding to ordinary and second sound, including the low-temperature relation between their velocities. The first approximation gives a one-particle Green function with finite sound absorption coefficients expressed through quantities characterizing the substance.
Full Text
UDC 532.132
PHYSICS
LI CHZHEN-CHZHUN
KINETIC EQUATIONS FOR SUPERFLUID BOSE SYSTEMS AND THEIR SOLUTIONS IN THE HYDRODYNAMIC APPROXIMATION
(Presented by Academician N. N. Bogolyubov on 9 IX 1965)
In the works of N. N. Bogolyubov \((^{1})\) and Z. Galiasevich \((^{2})\), asymptotic expressions for the Green functions for superfluid Bose systems were obtained with the aid of the equations of hydrodynamics. However, the coefficients and poles in these expressions for the Green functions remain uncalculated. The present work is devoted to the study of kinetic equations and their solutions in the hydrodynamic approximation for weakly interacting and weakly excited Bose systems. In the work, asymptotic expressions for the Green functions are found with already known coefficients and poles.
Let us take the Hamiltonian of the system in the form
\[ \mathcal H(\tau)=H+\delta H_\tau, \]
where \(\delta H_\tau\) corresponds to the introduction of external sources according to \((^{1})\); \(H\) is the Hamiltonian of a system of Bose particles with pair interaction. On the basis of our work \((^{3})\), it is convenient to express \(H\) in the representation of Bogolyubov quasiparticles as
\[ H=U+\sum_{f_1}\bigl(E(f_1)+\Delta(f_1)\bigr)\xi_{f_1}^{+}\xi_{f_1} +\varepsilon\sum_{f_1}\frac{1}{2}S_1(f_1)\bigl(\xi_{f_1}^{+}\xi_{-f_1}^{+}+\xi_{-f_1}\xi_{f_1}\bigr)+ \]
\[ +n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \sum_{f_1'f_2'f_1}Q(f_1',f_2';f_1) \bigl(\xi_{f_1'}^{+}\xi_{f_2'}^{+}\xi_{f_1} +\xi_{f_1}^{+}\xi_{f_2'}\xi_{f_1'}\bigr) \Delta(f_1'+f_2'-f_1)+\ldots, \tag{1} \]
where \(\varepsilon\) is a dimensionless small parameter \((^{4})\), and \(\xi_{f_1}^{+}, \xi_{f_1}\) are the creation and annihilation operators of quasiparticles. The remaining notation is given in \((^{3})\).
We define the two-time temperature Green functions for quasiparticles in the usual way:
\[ \langle\!\langle \xi_k(t);\xi_k^{+}(\tau)\rangle\!\rangle =-i\theta(t-\tau)\langle[\xi_k(t),\xi_k^{+}(\tau)]_-\rangle;\ \ldots \tag{2} \]
As is known, for these Green functions one can write a system of coupled equations of motion; for its approximate solution, approximate decouplings are necessary. In our case these decouplings have the form
\[ \langle\!\langle \xi_{f_1'}^{+}(t)\xi_{f_2'}^{+}(t)\xi_{f_2}(t);\xi_k^{+}(\tau)\rangle\!\rangle =\gamma_{f_1';f_2'}n_{f_1}\Delta(f_2-f_2') \langle\!\langle \xi_{f_1'}^{+}(t);\xi_k^{+}(\tau)\rangle\!\rangle+ \]
\[ +n_0^{1/2}(\varepsilon/V)^{1/2}c_1(f_1',f_2';f_2); \]
\[ \langle\!\langle \xi_{f_1}^{+}(t)\xi_{f_2'}^{-}(t)\xi_{f_2}^{+}(t);\xi_k^{+}(\tau)\rangle\!\rangle = n_0^{1/2}(\varepsilon/V)^{1/2}c_2(f_1,f_2',f_2); \tag{3} \]
\[ \langle\!\langle \xi_{f_1'}^{+}(t)\xi_{f_2'}^{+}(t)\xi_{f_2}(t)\xi_{f_1}(t);\xi_k^{+}(\tau)\rangle\!\rangle =\gamma_{f_1';f_2'}\gamma_{f_1;f_2}n_{f_1}\Delta(f_1-f_1') \langle\!\langle \xi_{f_2'}^{+}(t)\xi_{f_2}(t);\xi_k^{+}(\tau)\rangle\!\rangle; \]
\[ \langle\!\langle \xi_{f_1'}^{+}(t)\xi_{f_2'}^{+}(t)\xi_{f_2}^{+}(t)\xi_{f_1}(t);\xi_k^{+}(\tau)\rangle\!\rangle = \]
\[ =\frac{1}{2}\gamma_{f_1';f_2';f_2}n_{f_2}\Delta(f_1-f_1') \langle\!\langle \xi_{f_2'}^{+}(t)\xi_{f_2}(t);\xi_k^{+}(\tau)\rangle\!\rangle, \]
where
\[ n_f=\left(e^{E(f)/\theta}-1\right)^{-1},\qquad \gamma_{f_1;\ldots;f_s}=\sum_{(p)}(+1)^p \]
is the sum symmetrized over the indices. On the basis of this it is not difficult to obtain…
a closed system of equations of motion for the Green functions \(\langle\!\langle \xi_k(t);\xi_k^+(\tau)\rangle\!\rangle\), \(\langle\!\langle \xi_k^+(t);\xi_k^+(\tau)\rangle\!\rangle,\ldots,\langle\!\langle \xi_{f-k/2}(t)\xi_{f+k/2}^+(t);\xi_k^+(\tau)\rangle\!\rangle\). Since these equations are very cumbersome, we shall not give them here. Further, in deriving the kinetic equation we restricted ourselves to terms no higher than first order in the parameter \(\varepsilon\). In addition, all terms corresponding to multiple scatterings were discarded, and only diagonal matrix elements of three-quasiparticle processes were retained, in the sense of the work \({}^{(3)}\). Then in the hydrodynamic approximation (\(E,\mathbf{k}\) small quantities) we have the simplest form of the equations of motion for the Fourier transforms of the Green functions
\[ (E-E(k))\langle\!\langle \xi_k\mid \xi_k^+\rangle\!\rangle_E =\frac{1}{2\pi}+ \]
\[ +2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \sum_f Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)n_f(1+n_f)g(f,k\mid E), \tag{4} \]
\[ (E+E(k))\langle\!\langle \xi_{-k}^+\mid \xi_k^+\rangle\!\rangle_E= \]
\[ =-2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \sum_f Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)n_f(1+n_f)g(f,k\mid E), \]
where
\[ \langle\!\langle A_k\mid B_k\rangle\!\rangle_E =\frac{1}{2\pi}\int_{-\infty}^{\infty} \langle\!\langle A_k(t);B_k(\tau)\rangle\!\rangle e^{iE(t-\tau)}\,d(t-\tau). \]
The function
\[ g(f,k\mid E)=n_f^{-1}(1+n_f)^{-1} \langle\!\langle \xi_{f-k/2}^+\xi_{f+k/2}\mid \xi_k^+\rangle\!\rangle_E \]
satisfies the kinetic equation
\[ -\mathscr{D}(f,k\mid g)=\mathscr{L}_s(n_f\mid g), \tag{5} \]
where
\[ \mathscr{D}(f,k\mid g)=n_f(1+n_f)\{(-E+\mathbf{k}\cdot\mathbf{v}(f))g(f,k\mid E)+ \]
\[ +2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \mathbf{k}\cdot\mathbf{v}(f)\, Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right) \langle\!\langle \xi_k+\xi_{-k}^+\mid \xi_k^+\rangle\!\rangle_E, \tag{6} \]
\[ \mathscr{L}_s(n_f\mid g)=\frac{8\pi}{i}n_0\left(\frac{\varepsilon}{V}\right) \sum_{f_1',\,f_2'}\{|Q(f_1',f_2';f)|^2(1+n_{f_1'})(1+n_{f_2'})n_f\times \]
\[ \times\Delta(f_1'+f_2'-f)\delta(E(f_1')+E(f_2')-E(f))\times (g(f,k\mid E)-g(f_1',k\mid E)- \]
\[ -g(f_2',k\mid E))-2|Q(f,f_1';f_2')|^2(1+n_f)(1+n_{f_1'})n_{f_2'} \Delta(f+f_1'-f_2')\times \]
\[ \times\delta(E(f)+E(f_1')-E(f_2'))\times (g(f_2',k\mid E)-g(f_1',k\mid E)-g(f,k\mid E))\}, \tag{7} \]
where \(\mathscr{L}(t)=\vec{\nabla}_f E(f)\). Obviously, \(\mathscr{L}_s(n_f\mid g)\) is the collision integral, and \(\mathscr{D}(f,k\mid g)\) the drift terms of the kinetic equation.
Let us proceed to discuss the solutions of the kinetic equation (5) in the hydrodynamic approximation by means of the Chapman–Enskog method \({}^{(5)}\). We note that for \(\mathscr{L}_s(n_f\mid g)\) the relations
\[ 0=\int f_\alpha \mathscr{L}_s(n_f\mid g)\,d^3f =\int E(f)\mathscr{L}_s(n_f\mid g)\,d^3f. \]
hold. They correspond to the laws of conservation of quasiparticle momentum and energy in collisions. Taking these conservation laws into account, we may rewrite (5) in the form of “transport equations”:
\[ -Eb_1U_f^*(k\mid E)+\sum_\alpha k_\alpha \hat{P}_{j\alpha}(k\mid E)= \]
\[ =-2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \sum_\alpha k_\alpha A_1\delta_{j\alpha}k^{1/2} \langle\!\langle \xi_k+\xi_{-k}^+\mid \xi_k^+\rangle\!\rangle_E, \tag{8} \]
\[ -E\hat{\mathscr{E}}(k\mid E)+\sum_\alpha k_\alpha\hat{Q}_\alpha(k\mid E)=0, \]
in which the following notation has been used:
\[ \theta b_1 U_j^* (k|E)=\int f_j n_f(1+n_f)g(f,k|E)\,d^3f; \]
\[ \hat P_{j\alpha}(k|E)=\int f_j v_\alpha(f)n_f(1+n_f)g(f,k|E)\,d^3f; \tag{9} \]
\[ \hat{\mathcal E}(k|E)=\int E(f)n_f(1+n_f)g(f,k|E)\,d^3f; \]
\[ \hat Q_\alpha(k|E)=\int v_\alpha(f)E(f)n_f(1+n_f)g(f,k|E)\,d^3f, \]
where
\[ b_1=\frac{1}{3}\theta^{-2}\int f^2 n_f(1+n_f)\,d^3f; \]
\[ A_1=\int f_j v_j(t)k^{-1/2}Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)n_f(1+n_f)\,d^3f. \]
As in the work of B. I. Sadovnikov \((^6)\), we put
\(g(f,k|E)=g^{(0)}(f,k|E)+g^{(1)}(f,k|E)+\ldots\) and require that the equalities
\[ 0=\int f_\alpha n_f(1+n_f)g^{(1)}(f,k|E)\,d^3f =\int E(f)n_f(1+n_f)g^{(1)}(f,k|E)\,d^3f, \]
hold, i.e., that the quantities \(U_j^*(k|E)\), \(\hat{\mathcal E}(k|E)\) introduced by us be determined only through \(g^{(0)}(f,k|E)\).
From expression (7) it is easy to see that
\(0=\mathcal H_s(n_f|f_\alpha)=\mathcal H_s(n_f|E(f))\). Therefore, in the “acoustic approximation” \((^1)\) we may take \(g^{(0)}(f,k|E)\) in the form
\[ g^{(0)}(f,k|E)=\theta^{-1}\sum_\alpha f_\alpha U_\alpha^*(k|E) +\theta^{-2}E(f)\theta^*(k|E), \tag{10} \]
since the form \(g^{(0)}\) adopted by us satisfies the equation
\(0=\mathcal H_s(n_f|g^{(0)})\).
In the zeroth approximation, using expression (10) and definitions (9), we obtain
\[ \hat{\mathcal E}(k|E)=b_3\theta^*(k|E);\quad \hat Q_\alpha^{(0)}(k|E)=\theta b_2 U_\alpha^*(k|E);\quad \hat P_{j\alpha}^{(0)}(k|E)=\delta_{j\alpha}b_2\theta^*(k|E), \tag{11} \]
where
\[ b_2=\frac{1}{3}\theta^{-2}\int \mathbf f\cdot \mathbf v(f)E(f)n_f(1+n_f)\,d^3f, \qquad b_3=\theta^{-2}\int E^2(f)n_f(1+n_f)\,d^3f. \]
Substituting (11) into the transport equations (8), we find
\[ \theta^*(k|E)=2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \left(\frac{b_2A_1}{b_1b_3}\right) \frac{k^{5/2}}{E^2-\lambda^2k^2} \left\langle\!\left\langle \xi_k+\xi_{-k}^+ \mid \xi_k^+ \right\rangle\!\right\rangle_E; \]
\[ \sum_\alpha k_\alpha U_\alpha^*(k|E)= 2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2} \frac{k^{5/2}}{E^2-\lambda^2k^2} E\left\langle\!\left\langle \xi_k+\xi_{-k}^+ \mid \xi_k \right\rangle\!\right\rangle_E, \tag{12} \]
where \(\lambda^2=(b_2^2/b_1b_3)\). Thus, we have already found \(g^{(0)}(f,k|E)\), which is expressed in terms of the Green functions
\(\left\langle\!\left\langle \xi_k|\xi_k^+\right\rangle\!\right\rangle_E\),
\(\left\langle\!\left\langle \xi_k^+|\xi_k^+\right\rangle\!\right\rangle_E\).
Substituting this result into equation (4), we obtain
\[ \left\langle\!\left\langle a_k\mid a_k^+\right\rangle\!\right\rangle_E = -\frac{1}{2\pi} \frac{\Delta(k,E)n_0} {(E^2-c_0^2k^2)(E^2-c_1^2k^2)} = -\left\langle\!\left\langle a_{-k}^+\mid a_k^+\right\rangle\!\right\rangle_E, \tag{13} \]
where
\[ \lambda_1^2=(b_3A_1A_2/2\pi^3 b_1b_2); \qquad A_2=\theta^{-1}\int E(f)k^{-1/2}Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right) \]
\[ {}\times n_f(1+n_f)\,d^3f; \qquad c_{0,1}^2= \frac{1}{2k^2} \left\{(E^2(k)+\lambda^2k^2)\pm \left[(E^2(k)+\lambda^2k^2)^2 -4E^2(k)\lambda^2k^2 +8\varepsilon n_0\lambda_1^2E(k)k^3\right]^{1/2}\right\}. \]
In the low-temperature region \(\Delta(k,E)\) has the form \(\Delta(k,E)=\nu(0)\{-E^2+\lambda^2 k^2+\varepsilon n_0\lambda_1^2(n_0\nu(0)'/m)^{-1/2}k^2\}\). In this case all excitations are of the phonon type, and it is easy to estimate that \(c_1^2=\frac13 n_0\nu(0)'/m=\frac13 c_0^2\)—this is the well-known relation between ordinary and second sound near zero temperature.
In the first approximation the equation for \(g^{(1)}(f,k|E)\) has the form
\(-\mathscr{D}(f,k|g^{(0)})=\mathscr{L}_s(n_f|g^{(1)})\). Following the usual procedure, we express \(\mathscr{D}(f,k|g^{(0)})\) with the aid of the transport equations of the zeroth approximation and obtain the linearized integral equation for \(g^{(1)}\)
\[ \begin{aligned} n_f(1+n_f)\bigg\{& \theta^{-2}\sum_\alpha \left(v_\alpha(f)E(f)-\frac{b_2}{b_1}f_\alpha\right)k_\alpha\theta^*(k|E)+ \\ &+\theta^{-1}\sum_{\alpha,\beta}\left(f_\alpha v_\beta(f)-\frac{b_2}{b_3}E(f)\delta_{\alpha\beta}\right)k_\beta U_\alpha^*(k|E)+ \\ &+2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2}\sum_\alpha\left(k^{-1/2}Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)-\theta^{-2}\frac{A_1}{b_1}f_\alpha\right)\times \\ &\times k^{1/2}k_\alpha\langle\!\langle \xi_k+\xi_{-k}^{+}\mid \xi_k^{+}\rangle\!\rangle_E \bigg\} =\mathscr{L}_s(n_f|g^{(1)}). \end{aligned} \tag{14} \]
The solution of equation (14) may be represented in the form
\[ \begin{aligned} g^{(1)}(f,k|E)=i\bigg\{& \theta^{-2}\sum_\alpha\left(E(f)v_\alpha(f)-\frac{b_2}{b_1}f_\alpha\right)B_1k_\alpha\theta^*(k|E)+ \\ &+\theta^{-1}\sum_{\alpha,\beta}\left(f_\alpha v_\beta(f)-\frac{b_2}{b_3}E(f)\delta_{\alpha\beta}\right)B_2k_\beta U_\alpha^*(k|E)+2n_0^{1/2}\left(\frac{\varepsilon}{V}\right)^{1/2}\times \\ &\times\sum_\alpha\left(k^{-1/2}Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)-\theta^{-2}\frac{A_1}{b_1}f_\alpha\right)B_3k^{1/2}k_\alpha \langle\!\langle \xi_k+\xi_{-k}^{+}\mid \xi_k^{+}\rangle\!\rangle_E \bigg\}. \end{aligned} \tag{15} \]
For simplicity we regard \(B_1,B_2,B_3\) as constants. Then \(g^{(1)}(f,k|E)\) automatically satisfies the condition
\(0=\int f_\alpha n_f(1+n_f)g^{(1)}(f,k|E)\,d^3f=\int E(f)n_f(1+n_f)g^{(1)}(f,k|E)\,d^3f\), and it is easy to find the quantities \(B_1,B_2,B_3\) from equation (14). Using the expressions obtained in (10), (15) and the transport equations (8), by a method analogous to that used in the zeroth approximation, we obtain the one-particle Green function
\[ \langle\!\langle a_k\mid a_k^{+}\rangle\!\rangle_{E+i\varepsilon} \simeq -\frac{\Delta(k,E)n_0}{2\pi c_0c_1k^2(E+2i\varepsilon_0-c_0k)(E+2i\varepsilon_1-c_1k)}, \tag{16} \]
where
\[ \varepsilon_0=c_0^{-2}\left\{\frac12(E^2(k)-c_0^2k^2)\Gamma_1+\varepsilon n_0\lambda_1^2kE(k)\Gamma_2\right\}; \quad \varepsilon_1=c_0^{-2}\left\{\frac12E^2(k)\Gamma_1+\varepsilon n_0E_1kE(k)\Gamma_2\right\}; \]
\[ \Gamma_1=\left(\frac{b_2^2}{b_1b_3}-\frac{b_4}{b_3}\right)B_1 +2\left(\frac{b_2^2}{b_1b_3}-\frac{b_5}{b_1}\right)\frac{B_2}{\theta}; \quad \Gamma_2=\left(\frac{b_1A_3}{b_2A_1}-1\right)\theta B_2; \]
\[ b_4=\frac13\theta^{-2}\int v^2(f)E^2(f)n_f(1+n_f)\,d^3f; \quad b_5=\frac1q\theta^{-2}\int f^2v^2(f)n_f(1+n_f)\,d^3f; \]
\[ A_3=\frac13\theta^{-1}\int v^2(f)E(f)k^{-1/2}Q\left(k,f-\frac{k}{2};f+\frac{k}{2}\right)n_f(1+n_f)\,d^3f. \]
Comparing (16) with the result of Galyasevich\({}^{(2)}\), we see that now the absorption coefficients of first and second sound, \(\varepsilon_0,\varepsilon_1\), are already expressed in terms of known quantities related to the properties of the substance.
The author expresses profound gratitude to Academician N. N. Bogolyubov, Yu. A. Tserkovnikov, and V. A. Moskalenko for valuable advice and comments in carrying out this work.
Moscow State University
named after M. V. Lomonosov
Received
3 IX 1965
REFERENCES
- N. N. Bogolyubov, Preprint, Joint Institute for Nuclear Research, R-1395, 1963.
- Z. Galyasevich, Preprint, Joint Institute for Nuclear Research, R-1517, 1964.
- Li Tszyan-chazhun, DAN, 164, No. 6 (1965).
- Yu. A. Tserkovnikov, DAN, 143, No. 4 (1962).
- S. Chapman, T. Cowling, The Mathematical Theory of Non-Uniform Gases, 1960.
- B. I. Sadovnikov, DAN, 169, No. 1 (1966).