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UDC 530.145:536.75
PHYSICS
L. L. BUIISHVILI, M. D. ZVIADADZE
ON TAKING INTERACTION INTO ACCOUNT IN CONSTRUCTING A NONEQUILIBRIUM DENSITY MATRIX
(Presented by Academician N. N. Bogolyubov on 18 VII 1969)
In constructing a stationary nonequilibrium density matrix \((^{1})\) for a macrosystem consisting of several weakly coupled subsystems, ambiguity may arise in the question of the proper account of the interaction \(\mathscr{H}'\) between the subsystems. Usually this interaction is assigned the temperature of one of the subsystems \((^{2})\), or else it is noted that the final results do not depend on the assigned temperature \(\beta'\) so long as \(\mathscr{H}'\) can be regarded as a small perturbation \((^{3})\).
In the stationary case the energy conservation law holds
\[ \sum_{k=1}^{n}\frac{d\mathscr{H}_{k}}{dt}+\frac{d\mathscr{H}'}{dt}=0, \tag{1} \]
where \(\mathscr{H}_{k}\) is the Hamiltonian of the \(k\)-th subsystem; \(\dfrac{d\mathscr{H}_{k}}{dt}=\dfrac{1}{i}[\mathscr{H}_{k},\mathscr{H}]\), \(\mathscr{H}=\sum_{k}\mathscr{H}_{k}+\mathscr{H}'\) is the Hamiltonian of the system, and \(n\) is the number of subsystems.
It follows from (1) that when the system is divided into \(n\) subsystems only \(n-1\) are independent. In considering concrete processes the interaction energy is often not included in the energy balance and, instead of (1), the approximate conservation law \((^{3,4})\)
\[ \sum_{k=1}^{n}\frac{d\mathscr{H}_{k}}{dt}\simeq 0 \tag{2} \]
is used. Thus the term \(d\mathscr{H}'/dt\) is discarded; generally speaking, it is of the same order as the term \(\sum_{k=1}^{n} dH_{k}/dt\). It is not difficult to verify that, at least in the high-temperature approximation, neither \(\beta'\) nor \(d\mathscr{H}'/dt\) in (1) contributes to the mean values
\[ \overline{\mathscr{H}}_{k}=\operatorname{Sp}\rho \mathscr{H}_{k}, \qquad d\overline{\mathscr{H}}_{k}/dt=\operatorname{Sp}\rho\, d\mathscr{H}_{k}/dt, \]
which are used in deriving equations for the inverse temperatures of the subsystems. But sometimes one has to calculate the mean values \(\overline{A}_{j}=\operatorname{Sp}\rho A_{j}\) of such quantities \(A_{j}\) that are proportional to \(\mathscr{H}'\). In this case it is not evident in advance that the final result also will not depend on \(\beta'\), and that the use of (2) is legitimate.
In the present note this question is discussed for the example of calculating the real part of the complex susceptibility \(\chi\) for a spin system in external constant and alternating magnetic fields. The Hamiltonian of the system in the rotating coordinate system has the form
\[ \mathscr{H}=\mathscr{H}_{z}+\mathscr{H}_{d}+\mathscr{H}_{x}, \qquad \mathscr{H}_{z}=(\omega_{0}-\omega)S_{z}, \qquad \mathscr{H}_{x}=\omega_{1}S_{x}, \tag{3} \]
where \(\mathscr{H}_{d}\) is the secular part of the dipole–dipole interaction of the spins; \(\omega_{0}=\gamma H_{0}\); \(\omega_{1}=\gamma H_{1}\); \(H_{0}\) is the constant magnetic field; \(\omega, H_{1}\) are the frequency and
semiamplitude of the alternating field; \(S_\alpha=\sum_i S_{i\alpha}\), \(\alpha=x,y,z\), are spin operators.
According to (1), we form the generalized integrals of motion
\[ \begin{gathered} \widetilde{\mathscr H}_z=\mathscr H_z-\int_{-\infty}^{0} e^{\varepsilon t}\dot{\mathscr H}_z(t)\,dt,\qquad \widetilde{\mathscr H}_d=\mathscr H_d-\int_{-\infty}^{0} e^{\varepsilon t}\dot{\mathscr H}_d(t)\,dt,\\ \widetilde{\mathscr H}_x=\mathscr H_x-\int_{-\infty}^{0} e^{\varepsilon t}\dot{\mathscr H}_x(t)\,dt =\mathscr H_x+\int_{-\infty}^{0} e^{\varepsilon t}\{\dot{\mathscr H}_x(t)+\dot{\mathscr H}_d(t)\}\,dt . \end{gathered} \tag{4} \]
Let us note that, in contrast to works \((^{2-4})\), here \(\mathscr H_x\) is also introduced into the consideration. In the last expression (4) the exact conservation law has been used,
\[ \dot{\mathscr H}_x=-\dot{\mathscr H}_z-\dot{\mathscr H}_d . \tag{5} \]
The quasiequilibrium density matrix in the high-temperature approximation has the form
\[ \rho=\frac{1}{\operatorname{Sp}1}\left\{1-\beta_z\mathscr H_z-\beta_d\mathscr H_d-\beta_x\mathscr H_x +(\beta_z-\beta_x)\int_{-\infty}^{0} e^{\varepsilon t}\mathscr H_z(t)\,dt +(\beta_d-\beta_x)\int_{-\infty}^{0} e^{\varepsilon t}\dot{\mathscr H}_d(t)\,dt\right\}, \tag{6} \]
where \(\beta_z\) and \(\beta_d\) are the inverse temperatures of the Zeeman subsystem and the dipole-dipole reservoir, and \(\beta_x\) is the inverse temperature assigned to the invariant part of the interaction \(\widetilde{\mathscr H}_x\). The time dependence of the operators denotes the Heisenberg representation.
Using the density matrix (6) and a phenomenological treatment of spin-lattice relaxation, analogously to \((^{4,5})\), one easily obtains the equations
\[ \begin{gathered} \frac{d\beta_z}{dt}=-2W(\omega-\omega_0)(\beta_z-\beta_d) -\left(\beta_z-\frac{\omega_0}{\omega_0-\omega}\beta_L\right)\bigg/ T_z,\\ \frac{d\beta_d}{dt}=-2W(\omega-\omega_0)\frac{(\omega_0-\omega)^2}{\omega_d^2}(\beta_d-\beta_z) -\frac{\beta_d-\beta_L}{T_d},\\ \frac{d\beta_x}{dt}=-\frac{\beta_x-\beta_L}{T_x'}-\frac{\beta_x}{T_x''}, \end{gathered} \tag{7} \]
where \(W(\omega-\omega_0)=\dfrac{\pi\omega_1^2}{2}\,\varphi(\omega-\omega_0)\) is the usual probability of transitions induced by the alternating field; \(\varphi(\omega-\omega_0)\) is the absorption line shape; \(\omega_d^2=\operatorname{Sp}\mathscr H_d^2/\operatorname{Sp}S_z^2\); \(T_z\) and \(T_d\) are the times of spin-lattice relaxation of the Zeeman subsystem and the DDR; \(T_x'\) and \(T_x''\) are the relaxation times due respectively to the secular and nonsecular parts of the spin-lattice interaction \((^6)\); \(\beta_L\) is the inverse temperature of the lattice. It should be noted that the same equations for \(\beta_z\) and \(\beta_d\) are obtained also in the case when \(\widetilde{\mathscr H}_x\) is not taken into account and, instead of (5), the approximate conservation law is used,
\[ \dot{\mathscr H}_d+\dot{\mathscr H}_z=0 . \tag{8} \]
It follows from this that the inclusion of \(\widetilde{\mathscr H}_x\) also does not change the imaginary part of the complex susceptibility \(\chi''\). We shall now calculate \(\chi'\). We have \((^5)\)
\[ \chi'=M_x/2H_1, \tag{9} \]
where \(M_x\) is the mean value of the \(x\)-th component of the magnetization in the rotating coordinate system, which in our case is equal to
\[ M_x=\gamma \bar S_x=\gamma \operatorname{Sp}\rho S_x . \tag{10} \]
With the aid of (6) and (10) we obtain \((\langle\cdots\rangle \equiv \operatorname{Sp}(\cdots)/\operatorname{Sp}1)\)
\[ \bar S_x=-\beta_x\omega_1\langle S_x^2\rangle +(\beta_z-\beta_x)\int_{-\infty}^{0} e^{\varepsilon t} \langle S_x\dot{\mathcal H}_z(t)\rangle\,dt+ \]
\[ +(\beta_d-\beta_x)\int_{-\infty}^{0} e^{\varepsilon t} \langle S_x\dot{\mathcal H}_d(t)\rangle\,dt . \tag{11} \]
Evaluation of the integrals entering (12), to first order in \(\mathcal H_x\), gives
\[ \int_{-\infty}^{0} e^{\varepsilon t}\langle S_x\dot{\mathcal H}_z(t)\rangle\,dt = \langle S_x^2\rangle\,\omega_1(\omega-\omega_0)J_1(\omega-\omega_0), \]
\[ \int_{-\infty}^{0} e^{\varepsilon t}\langle S_x\dot{\mathcal H}_d(t)\rangle\,dt = -\langle S_x^2\rangle\,\omega_1 \left[1+(\omega-\omega_0)J_1(\omega-\omega_0)\right], \tag{12} \]
\[ J_1(x)=\int_{-\infty}^{\infty}\frac{\Phi(y)\,dy}{y-x}=-J_1(-x) \]
(the last integral \(\int_{-\infty}^{\infty}\) is taken in the sense of the principal value).
For \(\chi'\) we finally obtain
\[ \chi'=\frac{1}{2}\chi_0 \left\{ (\omega-\omega_0)J_1(\omega-\omega_0)\frac{\beta_z-\beta_d}{\beta_L} -\frac{\beta_d}{\beta_L} \right\}, \tag{13} \]
where \(\chi_0=N\gamma^2\langle S_z^2\rangle\beta_L\) is the static susceptibility. Thus, the terms with \(\beta_x\) do not enter into \(\chi'\), so that assigning a separate temperature to the interaction \(\mathcal H_x\), or arbitrarily including it in any of the subsystems, does not affect the final result. However, the use of the approximate expression (8) instead of the exact one (5) proves to be incorrect in finding \(\chi'\), since in this case the last term in (13) is lost.
The example considered shows that, at least in the lowest order of perturbation theory, taking account of the invariant part of the interaction \(\mathcal H'\) in constructing the nonequilibrium density matrix does not affect the expressions for \(\chi_x'\) and \(\chi_x''\). At the same time, the use of the approximate conservation law (2) may prove to be incorrect in calculating quantities proportional to the perturbation \(\mathcal H'\), as occurs for \(\chi'\). This latter circumstance must be taken into account when considering specific problems.
Received
11 VII 1969
CITED LITERATURE
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- D. N. Zubarev, Preprint ITF, 69-6, Kiev, 1969.
- R. Kubo, in: Thermodynamics of Irreversible Processes, IL, 1962.
- L. L. Buishvili, D. N. Zubarev, FTT, 7, 723 (1965); L. L. Buishvili, ZhETF, 49, 1868 (1965).
- B. N. Provotorov, ZhETF, 41, 1582 (1961).
- I. Solomon, J. Ezratty, Phys. Rev., 127, 78 (1962); N. S. Benyashvili, L. L. Buishvili, M. D. Zviadadze, Preprint ITF, 68-70, Kiev, 1968.