Full Text
D. N. ZUBAREV
STATISTICAL OPERATOR FOR NONSTATIONARY PROCESSES
(Presented by Academician N. N. Bogolyubov, January 26, 1965)
A statistical operator \(\rho\) for nonequilibrium stationary processes was constructed in \((^1)\) with the aid of local integrals of motion and applied to various problems \((^{2,3})\). We shall construct it by the method \((^{1,2})\) for nonstationary processes and apply it to the derivation of the equations of relativistic hydrodynamics \((^{4,5})\).
The possibility of constructing \(\rho\) for nonstationary processes is based on the fact that in irreversible processes there are two different time scales of relaxation—the time of rapid establishment of a state of local equilibrium and the time of its comparatively slow evolution \((^{6,7})\). Therefore, for not too rapid nonstationary processes, over a small interval of time there is established a nonstationary but locally equilibrium distribution, which depends on time only through its parameters and sufficiently slowly. We shall find this distribution by the method \((^{1,2})\).
In the nonstationary case \(\rho\) may be defined as the integral of the Liouville equation
\[ \frac{d\rho}{dt}=\frac{\partial \rho}{\partial t}+\frac{1}{i}[H,\rho]=0, \tag{1} \]
where the partial derivative denotes differentiation with respect to the time on which the parameters entering \(\rho\) explicitly depend, i.e. \(\partial \rho/\partial t\) represents the effect of an external action on the system that makes it nonstationary.
To find \(\rho\) for the nonstationary state of the system, we use a set of operators \(B_k(x,t)\), depending on the point \(x\) and on the time \(t\), and satisfying an equation of type (1),
\[ \frac{\partial B_k(x,t)}{\partial t}+\frac{1}{i}[H,B_k(x,t)]=0; \]
then, if \(\rho(t)\) is a functional of \(B_k(x,t)\), it satisfies (1), since the functional may be regarded as a function of the Fourier components of \(B_k(x,t)\) with respect to \(x\).
We shall proceed from the conservation laws of energy, particle number, and momentum, \(\dot P_k(x)+\nabla j_k(x)=0\) \((k=0,1,\ldots,l)\), where \(P_0(x)=H(x)\) is the energy density; \(P_i(x)=n_i(x)\) \((i=1,\ldots,l-1)\) is the density of the number of particles; \(P_l(x)=\mathbf{p}(x)\) is the momentum density; \(j_0(x)=j_H(x)\), \(j_i(x)=j_i(x)\), \(j_l(x)=T(x)\) are the densities of the fluxes of the corresponding quantities.
Let us construct the operators \(B_k(x,t)\), which depend on time through the parameters \(F_k(x,t)\) determining the macroscopic state of the system:
\[ B_k(x,t)=F_k(x,t)P_k(x)- \]
\[ -\int_{-\infty}^{0} e^{\varepsilon t_1} \left\{ F_k(x,t+t_1)\dot P_k(x,t_1) +\frac{\partial F_k(x,t+t_1)}{\partial t_1}P_k(x,t_1) \right\}\,dt_1, \tag{2} \]
where \(F_0(x,t)=\beta(x,t);\quad F_i(x,t)=-\beta(x,t)\left(\mu_i(x,t)-\frac{m_i}{2}v^2(x,t)\right)\).
\((i=1,2,\ldots,i-1);\ F_l(x,t)=-\beta(x,t)v(x,t);\ \beta(x,t)\) is the inverse temperature; \(\mu_i(x,t)\) is the chemical potential; \(v(x,t)\) is the mass velocity. The time argument of the operators \(P_k(x,t)\), \(\dot P_k(x,t)\) has a different meaning than in \(F_k(x,t)\), and denotes the Heisenberg representation. If \(F_k\) does not depend on time, then \(B_k(x,t)=F_k(x)A_k(x)\), where \(A_k(x)\) are the local integrals of motion considered in \((^{1,2})\). Operator (2) has the meaning of the invariant part of \(F_k(x,t+t_1)P_k(t_1)\) with respect to evolution in \(t_1\). Indeed, integrating (2) by parts, we obtain
\[ B_k(x,t)=\varepsilon\int_{-\infty}^{0} e^{\varepsilon t_1}F_k(x,t+t_1)P_k(x,t)\,dt_1. \]
The limiting transition \(\varepsilon\to0\) should be performed after letting the volume of the system tend to infinity.
Following \((^{1,2})\), we construct the statistical operator
\[ \rho(t)=Q^{-1}\exp\left\{-\sum_k\int B_k(x,t)\,dx\right\} =Q^{-1}\exp\left\{-\sum_k\int\left(F_k(x,t)P_k(x)-\right.\right. \]
\[ \left.\left. -\int_{-\infty}^{0} e^{\varepsilon t_1}\left((\nabla F_k(x,t+t_1))j_k(x,t_1) +\frac{\partial F_k(x,t+t_1)}{\partial t_1}P_k(x,t_1)\,dt_1\right)dx\right\}, \tag{3} \]
where we have used the conservation equations and have omitted surface integrals.
In the stationary case (3) coincides with the operator obtained in \((^1)\), and in the nonstationary case with the result of MacLennan \((^8)\). It is easy to show that (3) satisfies equation (1) as \(\varepsilon\to0\). In introducing the statistical operator (3), which depends on the parameters \(F_k(x,t)\), we assume that they are sufficient for describing the macroscopic state of the system and that the fluctuations \(P_k(x)\) are not too large.
Operator (3) makes it possible to study the transport of mechanical quantities in the nonstationary case. Taking the thermodynamic forces \(\nabla F_k,\ \partial F_k/\partial t\) to be small, we obtain linear relations between the fluxes and the thermodynamic forces:
\[ \langle j_k(x)\rangle=\langle j_k(x)\rangle_l+ \sum_m\int\!\!\int_{-\infty}^{0} e^{\varepsilon t_1} \left\{(j_k(x),j_m(x',t_1))\nabla F_m(x',t+t_1)+\right. \]
\[ \left. +(j_k(x),P_m(x',t_1))\frac{\partial F_m(x',t+t_1)}{\partial t}\right\}\,dx'\,dt_1, \tag{4} \]
where the brackets \((j_k,j_m)\) are correlation functions of the type
\[ (j_k,j_m)=\beta^{-1}\int_{0}^{\beta} \langle j_k(j_m(i\tau)-\langle j_m\rangle)\rangle_l\,d\tau; \tag{5} \]
\[ j_m(i\tau)=e^{-B\tau}j_m e^{B\tau}; \]
\(\langle\ldots\rangle_l\) denotes the average over the locally equilibrium state.
Let the thermodynamic forces vary periodically in time,
\[ F_k(x,t)=\sum_{\omega}F_k(x,\omega)e^{i\omega t}, \]
then
\[ \langle j_k(x)\rangle=\langle j_k(x)\rangle_l+ \tag{6} \]
\[ +\sum_{m,\omega}e^{i\omega t}\int\left\{(j_k(x),j_m(x'))_{\omega}\nabla F_m(x',\omega) +(j_k(x),P_m(x'))_{\omega}i\omega F_m(x',\omega)\right\}\,dx', \]
where
\[ (A,B)_{\omega}=\int_{-\infty}^{0}e^{\varepsilon t+i\omega}(A,B)\,dt \]
are kinetic coefficients depending on the frequency. Thus, in addition to spatial dispersion, there is also frequency dispersion of the kinetic coefficients.
The linear relations (4) can be written for an isotropic medium by separating scalar, vector, and tensor processes. As an example,
Consider the equations of relativistic hydrodynamics \((^{4,5})\). We shall proceed from the law of conservation of energy–momentum
\[ \sum_{\mu=1}^{4}\frac{\partial T_{\mu\nu}(x,t)}{\partial x_\mu}=0 \qquad (x_4=ict), \tag{7} \]
where \(T_{\mu\nu}\) is the symmetric operator tensor of the energy–momentum density. The conservation law (7) corresponds to the statistical operator
\[ \rho=Q^{-1}\exp\left\{\sum_\nu\int F_\nu(x,t)T_{4\nu}(x)\,dx-\right. \]
\[ \left. -\sum_{\mu\nu}\int_{-\infty}^{0}\!\!\int e^{\varepsilon t_1}ic\, \frac{\partial F_\nu(x,t+t_1)}{\partial x_\mu}\, T_{\mu\nu}(x,t_1)\,dx\,dt_1 \right\}. \tag{8} \]
The parameters \(F_\nu(x,t)\) are related to the four-dimensional velocity \(u_\nu\) and the invariant temperature \(\beta^{-1}\):
\[ F_\nu(x,t)=-\beta(x,t)iu_\nu(x,t). \]
Consider a locally equilibrium state, putting \(\partial F_\nu/\partial x_\mu=0\). Then, in the moving system \(u_1'=u_2'=u_3'=0,\ u_4'=i\), the statistical operator (8) has the usual, nonrelativistic form, which confirms the correctness of our definition. Choose \(u_\nu\) so that \(\ln Q_l\) is extremal with respect to \(u_\nu\), i.e. \(\delta\ln Q_l/\delta u_\nu(x)=0\); then we obtain \(u_\nu(x)/u_4(x)=\langle T_{4\nu}(x)\rangle/\langle T_{44}(x)\rangle=iv_\nu(x)/c,\quad u_\nu(x)=v_\nu(x)/c\sqrt{1-v^2(x)/c^2}\), \(u_4(x)=i/\sqrt{1-v^2(x)/c^2}\), i.e. the well-known relativistic relations for local quantities. The meaning of the parameter \(\beta\) as the reciprocal of the “proper” temperature is evident from the thermodynamic equality
\[ \delta\ln Q_l/\delta\beta(x)=\langle T_{44}(x)\rangle\sqrt{1-v^2(x)/c^2}. \tag{9} \]
With the aid of (8) we obtain linear relations between the mean energy–momentum tensor and the thermodynamic forces \(\partial F_{\nu_1}/\partial x_{\mu_1}\), assuming the latter to be small:
\[ \langle T_{\mu\nu}(x)\rangle=\langle T_{\mu\nu}(x)\rangle_l- \]
\[ -\sum_{\mu_1\nu_1}\int_{-\infty}^{0}\!\!\int e^{\varepsilon t_1} \bigl(T_{\mu\nu}(x),T_{\mu_1\nu_1}(x',t_1)\bigr)ic\, \frac{\partial F_{\nu_1}(x',t+t_1)}{\partial x'_{\mu_1}}\,dx'\,dt_1. \tag{10} \]
Expression (10) unites all irreversible processes in a system with the conservation law (7). For an isotropic medium it is convenient to separate in it processes of different tensor dimensionality, decomposing the tensor \(T_{\mu\nu}\) into proper components relative to the hydrodynamic motion
\[ T_{\mu\nu}=\varepsilon u_\mu u_\nu+P_\mu u_\nu+P_\nu u_\mu+P_{\mu\nu}, \tag{11} \]
where
\[ \sum_\mu P_\mu u_\mu=\sum_\nu P_{\mu\nu}u_\nu=0;\quad \varepsilon=\sum_{\mu\nu}u_\mu T_{\mu\nu}u_\nu;\quad P_\mu=cq_\mu=-\sum_{\nu\lambda}\Delta_{\mu\nu}T_{\nu\lambda}u_\lambda;\quad P_{\mu\nu}= \]
\[ =\sum_{\mu_1\nu_1}\Delta_{\mu\mu_1}T_{\mu_1\nu_1}\Delta_{\nu_1\nu}; \quad \varepsilon \text{ is the proper energy;} \quad q_\mu \text{ is the heat flux;} \quad P_{\mu\nu} \text{ is the} \]
stress tensor; \(\Delta_{\mu\nu}=\delta_{\mu\nu}+u_\mu u_\nu\). Introduce the tensor of viscous shear stresses \(\pi_{\mu\nu}\)
\[ P_{\mu\nu}=\pi_{\mu\nu}+p\Delta_{\mu\nu},\qquad p=\frac{1}{3}\sum_\mu P_{\mu\mu}, \tag{12} \]
where \(p\) is the pressure operator.
Using Curie’s theorem, one can separate, for an isotropic medium, the tensor, vector, and scalar processes:
\[ \langle \pi_{\mu\nu}(x)\rangle = -\sum_{\mu_1\nu_1}\int\!\!\int_{-\infty}^{0} e^{\varepsilon t_1} \bigl(\pi_{\mu\nu}(x),\pi_{\mu_1\nu_1}(x',t_1)\bigr) c\beta(x',t+t_1) \frac{\partial u_\nu(x',t+t_1)}{\partial x'_{\nu_1}} \,dx'\,dt_1, \]
\[ \begin{aligned} \langle P_\mu(x)\rangle &=c\langle q_\mu(x,t)\rangle =\sum_\nu\int\!\!\int_{-\infty}^{0} e^{\varepsilon t_1} \bigl(P_\mu(x),P_\nu(x',t_1)\bigr)\times \\ &\quad \times \left( c\frac{\partial\beta(x',t+t_1)}{\partial x'_\nu} -\beta(x',t+t_1)D u_\nu(x',t+t_1) \right) \,dx'\,dt_1, \end{aligned} \tag{13} \]
\[ \begin{aligned} \langle p(x)\rangle-\langle p(x)\rangle_l &= -\int\!\!\int_{-\infty}^{0} e^{\varepsilon t_1} \Bigl\{ \bigl(p(x),p(x',t_1)\bigr)c\beta(x',t+t_1)\operatorname{div}\mathbf{u}(x',t+t_1) \\ &\quad -\bigl(p(x),\varepsilon(x',t_1)\bigr)D\beta(x',t+t_1) \Bigr\} \,dx'\,dt_1, \end{aligned} \]
where
\[ D=c\sum_\mu u_\mu\frac{\partial}{\partial x_\mu}. \]
From (13), if spatial and temporal dispersion is neglected, equations \(({}^{4,5,9})\) follow. The method presented is readily generalized to the case where, in addition to energy and momentum, it is necessary to take into account the conservation of charges of various types, the angular momentum, and the difference in the velocities and temperatures of the components of a liquid, which may be of interest in the theory of multiple particle production in collisions of fast nucleons with nuclei \(({}^{9,10})\).
In conclusion I express my gratitude to N. N. Bogolyubov for a valuable discussion.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
20 I 1965
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