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UDC 533.7
PHYSICS
I. P. Pavlotskii
THE KINETIC EQUATION IN THE PRESENCE OF NON-LAGRANGIAN FORCES
(Presented by Academician N. N. Bogolyubov, July 3, 1969)
The aim of the present work is to obtain an analogue of the Boltzmann equation for a system of \(N\) particles \((N \to \infty)\), whose coordinates \(q_j\) and momenta \(p_j\) obey the quasicanonical equations:
\[ \dot q_j=\partial H/\partial p_j;\qquad \dot p_j=-\partial H/\partial q_j+Q_j(p_j);\qquad j=1,2,\ldots,N. \tag{1} \]
Here \(Q_j\) are non-Lagrangian forces, an example of which may be friction; \(H\) is the Hamiltonian function of the system obtained from the one under consideration if one sets \(Q_j=0\) \((j=1,2,\ldots,N)\).
As shown in works \((^{2,4})\), for the \(s\)-particle correlation functions \(F_s\) the recurrence equations are valid:
\[ \frac{\partial F_s}{\partial t} = \{H_s;F_s\} - \sum_{(1\leq j\leq s)} \frac{\partial}{\partial p_j}(Q_j\cdot F_s) + \frac{1}{v}\int dx_{s+1} \left\{ \sum_{(1\leq j\leq s)} \Phi_{j,s+1};F_{s+1} \right\}. \tag{2} \]
Here the braces are Poisson brackets; \(v=\lim(V/N)\) as \(V,N\to\infty\); \(V\) is the volume occupied by the system in configuration space; \(\Phi_{jk}\) is the interaction potential between the \(j\)-th and \(k\)-th particles. For brevity of notation we denote \(x_j=(p_j,q_j)\), \(t\) is time, and \(H_s\) is the Hamiltonian function of a system of \(s\) particles.
To obtain the kinetic equation from system (2), we shall use an analogue of N. N. Bogolyubov’s reasoning \((^1)\). It can be shown that in the case under consideration all the assumptions sufficient for the evolution of the function \(F_1(x_1,t)\) at the kinetic stage to be described by the equation
\[ \partial F_1/\partial t=A(x_1|F_1), \tag{3} \]
where \(A\) is a quantity functionally dependent on \(F_1\) and not depending explicitly on time, are valid. It is also assumed that for \(s\geq 2\), \(F_s=F_s(x_1,\ldots,x_s|F_1)\). Put
\[ A=\sum_{(0\leq j<\infty)} v^{-j}A_j(x_1|F_1);\qquad F_s=\sum_{(0\leq j<\infty)} v^{-j}F_s^{(j)}. \tag{4} \]
Substituting (4) into (2) for \(s=1\) and into (3), and then equating the coefficients of like powers of \(v\), we obtain:
\[ A_0=-\left(\frac{p_1}{m}\frac{\partial}{\partial q_1} + \frac{\partial}{\partial p_1}Q_1\right)F_1;\qquad A_k=\int dx_2\{\Phi_{12};F_2^{(k-1)}\}\quad(k>0). \tag{5} \]
Introduce \(D^{(r)}\), the operator of forming the derivative with respect to \(t\) followed by the substitution of \(\partial F_1/\partial t\) by \(A_r\). In this case the arguments of \(A_r\) and \(\partial F_1/\partial t\) are assumed to coincide. Comparing the expression for \(\partial F_s/\partial t\) obtained by substituting into the right-hand side of (2) the virial expansions (4) of the functions \(F_s\) and \(F_{s+1}\), and the expression for \(\partial F_s/\partial t\) written with the aid of the virial expansion (4) of \(F_s\) and the intro-
operator \(D^{(r)}\), we obtain functional equations for \(F_s^{(k)}\):
\[ D^0 F_s^0=\{H_s;F_s^0\}- \sum_{(1\leqslant j\leqslant s)} \frac{\partial}{\partial p_j}(Q_j(p_j)\cdot F_s^0); \]
\[ D^{(1)}F_s^0+D^0F_s^{(1)}=\{H_s;F_s^{(1)}\}- \sum_{(1\leqslant j\leqslant s)} \frac{\partial}{\partial p_j}(Q_j\cdot F_s^{(1)})+ \]
\[ +\int dx_{s+1}\left\{ \sum_{(1\leqslant j\leqslant s)} \Phi_{j,s+1};F_{s+1}^0 \right\}; \tag{6} \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
A detailed derivation of equations (6) is given in the preprint \((^4)\). The first of equations (6) are closed equations determining the set \(\{F_s^0\}\). If \(\{F_s^0\}\) is known, the second of equations (6) determine the set \(\{F_s^{(1)}\}\), and so on. Thus, all equations (6) have the form
\[ D^0F_s^{(k)}=\{H_s;F_s^{(k)}\}- \sum_{(1\leqslant j\leqslant s)} \frac{\partial}{\partial p_j}(Q_jF_s^{(k)})-\Psi_s^{(k)}, \tag{6a} \]
where \(\Psi_s^{(k)}\) are known functionals of \(F_1\).
Introduce the \(k\)-particle evolution operator \((^3)\)
\[ S_k^*(t)=\exp\left[ t\sum_{(1\leqslant j\leqslant k)} \left( \frac{\partial H_k}{\partial q_j}\frac{\partial}{\partial p_j} -\frac{\partial H_k}{\partial p_j}\frac{\partial}{\partial q_j} +\frac{\partial}{\partial p_j}Q_j \right) \right], \tag{7} \]
which coincides with the evolution operator of the Lagrangian system if in (7) one sets \(Q_j=0\). It has the properties \((^3)\)
\[ S_k^*(0)=1,\quad S_k^*(t')S_k^*(t'')=S_k^*(t'+t''). \tag{8} \]
As is easy to see,
\[ \frac{\partial}{\partial \tau}S_1^*(\tau)F_1(x_1,t)= \left(\frac{p_1}{m}\frac{\partial}{\partial q_1}+ \frac{\partial}{\partial p_1}Q_1\right)S_k^*(\tau)F_1(x_1,t) =-A_0(x_1\mid S_1^*(\tau)F_1). \tag{9} \]
From formula (9) and the definition of \(D^0\) it follows that
\[ D^0F_s^{(k)}(x_1,\ldots,x_s\mid S_1^*(\tau)F_1)=-\partial F_s^{(k)}/\partial\tau. \tag{10} \]
In view of the arbitrariness of \(F_1\), replace in (6a) \(F_1\) by \(S_1^*(\tau)F_1\), after which, with the aid of (10), we arrive at the equations:
\[ \frac{\partial F_s^{(k)}}{\partial\tau} =\{H_s;F_s^{(k)}\}+ \sum_{(1\leqslant j\leqslant s)} \frac{\partial}{\partial p_j}(Q_j\cdot F_s^{(k)}) -\Psi_s^{(k)}(x_1,\ldots,x_s\mid S_1^*(\tau)F_1); \]
\[ F_s^{(k)}=F_s^{(k)}(x_1,\ldots,x_s\mid S_1^*(\tau)F_1). \tag{11} \]
As boundary conditions we take the principle of weakening of correlations in the remote past:
\[ \lim_{\tau\to\infty}S_m^*(-\tau)F_m(x_1,\ldots,x_m\mid S_1^*(\tau)F_1) = \lim_{\tau\to\infty}S_m^*(-\tau) \prod_{(1\leqslant j\leqslant m)} S_1^*(\tau)F_1(x_j,t), \tag{12} \]
or, by virtue of (4):
\[ \text{a) }\quad \lim_{\tau\to\infty}S_m^*(-\tau)F_m^0(x_1,x_2,\ldots,x_m\mid S_1^*(\tau)F_1) = \]
\[ = \lim_{\tau\to\infty}S_m^*(-\tau) \prod_{(1\leqslant j\leqslant m)} S_1^*(\tau)F_1(x_j,t); \]
\[ \text{b) }\quad \lim_{\tau\to\infty}S_m^*(-\tau)F_m^{(k)}(x_1,\ldots,x_m\mid S_1^*(\tau)F_1)=0 \quad (k>0). \tag{13} \]
It can be shown that for a system with monotonically decreasing energy (for example, under the action of friction forces), the weakening of correlations between the dynamical states of particles in the remote past occurs more effectively than in a system whose energy is constant. When considering a system that increases its energy due to the action of forces \(Q_j\), we shall assume an adiabatic inclusion of the forces \(Q_j\) at that stage of the system’s development when correlations were absent.
The solution of equation (11) for \(k=0\) has the form
\[ F_m^0(x_1,\ldots,x_m\mid S_1^*(\tau)F_1) = S_m^*(\tau)F_m^0(x_1,\ldots,x_m\mid F_1). \tag{14} \]
Acting on both sides of (14) with the operator \(S_m^*(-\tau)\), taking into account the properties (8), and then passing to the limit as \(\tau\to\infty\), we obtain, on the basis of (13),
\[ F_m^0(x_1,\ldots,x_m\mid F_1) = T_m F_1(x_1,t)\ldots F_1(x_m,t); \]
\[ T_m=\lim_{\tau\to\infty} S_m^*(x_1,\ldots,x_m,-\tau) \prod_{(1\le j\le m)} S_1^*(x_j,\tau). \tag{15} \]
Let us write the kinetic equation of the first approximation:
\[ \frac{\partial F_1}{\partial t} = -\left[ \frac{p_1}{m}\frac{\partial}{\partial q_1} + \frac{\partial}{\partial p_1}Q_1(p_1) \right]F_1 + \frac{1}{v}A_1(x_1,F_1); \]
\[ A_1= \int dx_2\{\Phi_{12};F_2^0(x_1,x_2\mid F_1)\} = \int dx_2\{\Phi_{12};T_2F_1(x_1,t)F_1(x_2,t)\}. \tag{16} \]
Denote by \(X_1(x_1,x_2,t)\) and \(X_2(x_1,x_2,t)\) the sets of coordinates and momenta that the first and second particles, respectively, would have at time \(t\) if, beginning from the “infinite past,” they interacted only with each other (and were also subjected to the action of the forces \(Q_{1,2}\)).
Then
\[ \{\Phi_{12};T_2F_1(x_1,t)F_1(x_2,t)\} = \{\Phi_{12};F_1(X_1)F_1(X_2)\}. \tag{17} \]
Using the first of equations (6), relation (17), and also the definition of the operator \(D^0\) and formula (5) for the operator \(A_0\), after standard transformations analogous to those carried out in the monograph \((^1)\), we arrive at the following expression for \(A_1\):
\[ A_1=I_B+\int dx_2 \left[ Q(p_1)F_1(X_1)\frac{\partial}{\partial p_1}F_1(X_2) + Q(p_2)F_1(X_2)\frac{\partial}{\partial p_2}F_1(X_1) \right], \tag{18} \]
where \(I_B\) is the well-known expression for the Boltzmann collision integral \((^1)\). The integration in (18) is over the region of interaction of the particles in configuration space and over the layer inside the sphere \(\vec p=p_{20}\) and \(\vec p=p_{21}\) in momentum space. We have denoted by \(p_{20}\) and \(p_{21}\) the momenta of the second particle, respectively, at the beginning and at the end of the interaction.
Let us note that outside the sphere of interaction
\[ \partial F_1(X_2)/\partial \vec p_1=0,\qquad \partial F_1(X_1)/\partial \vec p_2=0, \tag{19} \]
since in this case \(X_1=X_1(x_1)\) and \(X_2=X_2(x_2)\). Consequently, the integrand in formula (18) vanishes outside the region of interaction.
On the basis of (16) and (18), the final form of the kinetic equation of the first approximation is as follows:
\[ \frac{\partial F_1}{\partial t} + \frac{\vec p_1}{m}\frac{\partial F_1}{\partial q_1} + \frac{\partial}{\partial p_1}\bigl(Q(p_1)F_1\bigr) = \frac{1}{v} \left[ I_B+\int dx_2 \left[ Q(p_1)F_1(X_1)\frac{\partial}{\partial p_1}F_1(X_2) + Q(p_2)F_1(X_2)\frac{\partial}{\partial p_2}F_1(X_1) \right] \right]. \tag{20} \]
If during the interaction the principal change in momenta occurs due to the action of intermolecular forces, and not the forces \(Q_j\), then the last term on the right-hand side of (20) may be neglected in comparison with \(I_B\).
In other words, if \(Q_j=\alpha f(p_j)\), where \(\alpha\) is a small dimensionless parameter, then in the equation of the first approximation one may neglect the term preceded by the product of two small parameters—\((\alpha/\nu)\).
In conclusion, I express my gratitude to N. N. Bogolyubov for discussing the work.
Institute of Applied Mathematics
Academy of Sciences of the USSR
Moscow
Received
25 IV 1969
REFERENCES
- N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, Moscow, 1946.
- I. P. Pavlotskii, DAN, 182, No. 4, 799 (1968).
- I. P. Pavlotskii, Problems of the Statistical Mechanics of Non-Lagrangian Systems. I. Formal Equations of the Liouville Hierarchy, Preprint, Institute of Applied Mathematics, Academy of Sciences of the USSR, No. 11, 1969.
- I. P. Pavlotskii, Problems of the Statistical Mechanics of Non-Lagrangian Systems. II. Bogolyubov Equations for Correlation Functions, Preprint, Institute of Applied Mathematics, Academy of Sciences of the USSR, No. 18, 1969.
- I. P. Pavlotskii, Problems of the Statistical Mechanics of Non-Lagrangian Systems. III. Kinetic Equation for Classical Gases, Preprint, Institute of Applied Mathematics, Academy of Sciences of the USSR, No. 23, 1969.