PHYSICS

I. P. BAZAROV

ON THE VLASOV EQUATION FOR A CRYSTAL

(Presented by Academician N. N. Bogolyubov, 12 XI 1962)

In a number of works on the theory of the crystalline state, based on the equation with a self-consistent field of classical statistical physics \((^{1-3})\), the main attention was concentrated on the analysis of the emergence of a crystal from a spatially homogeneous distribution of particles as a result of the instability of such a distribution. Along these lines a crystallization criterion was obtained and the temperature of the phase transition was calculated for certain substances. The general shortcoming of such an approach to the theory of the crystal is that the equation with a self-consistent field is not a good approximation in the case of a homogeneous density distribution. This equation is most effective for large spatial inhomogeneity; in the case of small inhomogeneity its applicability is restricted by the long-range forces of interaction between particles.

In the present note we wish to show that the linearized kinetic Vlasov equation with a self-consistent field for variations of the density of a crystal at temperature \(T = 0\) gives not only the well-known equations of small oscillations, which determine the complete Born spectrum of the eigenfrequencies of a crystal, but that this linear spectrum is the only one for it.

Let \(\Phi(\mathbf r)\) be the potential energy of interaction between particles. The equilibrium phase density of the particle distribution, determined from the minimum of the potential energy of the system, is expressed by the function

\[ f_0(\mathbf r,\mathbf p)=\sum_i \delta(\mathbf r-\mathbf r_i)\delta(\mathbf p), \]

where

\[ \sum_i \nabla_{\mathbf r}\Phi(|\mathbf r-\mathbf r_i|)=0 \]

for each site of the crystal lattice \(\mathbf r=\mathbf r_j\). For a small deviation of the crystal density from the equilibrium value, the particle distribution function is equal to

\[ f(\mathbf r,\mathbf p,t)=f_0(\mathbf r,\mathbf p)+\varphi(\mathbf r,\mathbf p,t), \]

where the variation of the phase density \(\varphi(\mathbf r,\mathbf p,t)\) is determined by the kinetic Vlasov equation

\[ \frac{\partial \varphi}{\partial t}+\frac{1}{m}(\mathbf p,\nabla_{\mathbf r}\varphi) =(\nabla_{\mathbf r}U_0,\nabla_{\mathbf p}\varphi)+(\nabla_{\mathbf r}U,\nabla_{\mathbf p}f_0), \tag{1} \]

in which

\[ U_0(r)=-\int \Phi(|\mathbf r-\mathbf r'|)f_0(\mathbf r',\mathbf p')\,d\mathbf r'\,d\mathbf p' =\sum_i \Phi(|\mathbf r-\mathbf r_i|), \]

\[ U(r)=\int \Phi(|\mathbf r-\mathbf r'|)\varphi(\mathbf r',\mathbf p',t)\,d\mathbf r'\,d\mathbf p'. \]

Substituting into (1) the expression

\[ \varphi(\mathbf r,\mathbf p,t) =-\sum_i(\nabla_{\mathbf r}\delta(\mathbf r-\mathbf r_i),\mathbf Q_i(t))\delta(\mathbf p) -\sum_i(\nabla_{\mathbf p}\delta(\mathbf p),\mathbf P_i(t))\delta(\mathbf r-\mathbf r_i), \tag{2} \]

we obtain

\[ \sum_i \left( \nabla_r \delta(\mathbf r-\mathbf r_i),\ \frac{d\mathbf Q_i}{dt}\right)\delta(\mathbf p) +\sum_i \left( \nabla_p \delta(\mathbf p),\ \frac{d\mathbf P_i}{dt}\right)\delta(\mathbf r-\mathbf r_i) - \]

\[ -\frac{1}{m}\sum_i \left( \nabla_r\delta(\mathbf r-\mathbf r_i),\ \mathbf P_i\right)\delta(\mathbf p) = \sum_{i,j}\left(\nabla_p\delta(\mathbf p),\ \nabla_r\bigl(\nabla_r\Phi(|\mathbf r-\mathbf r_j|),\ \mathbf Q-\mathbf Q_i\bigr)\right)\delta(\mathbf r-\mathbf r_i), \]

whence it is seen that (2) is a solution of equation (1), if \(\mathbf Q_i\) and \(\mathbf P\) satisfy the equations:

\[ m\frac{d\mathbf Q_i}{dt}=\mathbf P_i,\qquad \frac{d\mathbf P_i}{dt}= \nabla_{\mathbf r_i}\sum_j \bigl(\nabla_{\mathbf r_i}\Phi(|\mathbf r_i-\mathbf r_j|),\ \mathbf Q_j-\mathbf Q_i\bigr). \tag{3} \]

But the system of equations (3) determines the small oscillations of the particles of a crystal lattice \({}^{(4)}\), so that

\[ \mathbf r_i(t)=\mathbf r_i+\mathbf Q_i(t),\qquad \mathbf p_i(t)=\mathbf P_i(t). \]

Since the variation of the density \(\rho(\mathbf r,t)=\int \varphi(\mathbf r,\mathbf p,t)\,d\mathbf p\) is a linear function of \(\mathbf Q_i\), it therefore oscillates with the same frequencies.

Thus, the Born spectrum of the frequencies of the natural oscillations of a crystal lattice is completely determined by the Vlasov kinetic equation for the variation of the phase density.

In the particular case of a one-dimensional lattice model with period \(a\), consisting of alternating particles with masses \(m\) and \(M\), and taking into account interaction only with neighboring particles, from (3) we directly obtain the system of equations

\[ m\ddot Q_i=\alpha(Q_{i+1}-Q_{i-1}-2Q_i), \]

\[ M\ddot Q_{i+1}=\alpha(Q_{i+2}-Q_i-2Q_{i+1}) \qquad (\alpha=\Phi''(a)), \]

which leads to the well-known acoustic and optical oscillations of the lattice in this case \({}^{(4)}\).

The result obtained can also be established by proceeding from the fact that the nonlinear Vlasov equation (1) has a particular solution of the form

\[ f(\mathbf r,\mathbf p,t)=\sum_i \delta(\mathbf r-\mathbf r_i(t))\delta(\mathbf p-\mathbf p_i(t)), \]

if \(\mathbf r_i(t),\mathbf p_i(t)\) are determined from the equations of mechanics in the approximation of small oscillations. However, the existing prejudice concerning the inapplicability of this equation to a crystal hindered the establishment of the result obtained, as a consequence of which this result itself appears somewhat unexpected, although it could have been obtained directly more than 10 years ago*.

Let us now show that equation (1) not only possesses, among other possibilities, the Born spectrum, but that this spectrum is the only one for it. To this end we adiabatically turn on in the system described by equation (1) an infinitely small external periodic action

\[ W=\sum_{\Omega} V_{\Omega}(\mathbf r)e^{i\Omega t+\varepsilon t} \qquad (\varepsilon\to 0,\ \varepsilon>0), \tag{4} \]

so that

\[ \frac{\partial\varphi}{\partial t} +\frac{1}{m}(\mathbf p,\nabla_r\varphi) -(\nabla_r U_0,\nabla_p\varphi) -(\nabla_r U,\nabla_p f_0) =F(t,\mathbf r,\mathbf p), \]

\[ F(t,\mathbf r,\mathbf p)=(\nabla_r W,\nabla_p f_0), \tag{5} \]

* Interesting considerations on this question may be found in \({}^{(5)}\).

and set, as always, \(\varphi=0\) for \(t=-\infty\). Under this initial condition, expression (2) is a solution of equation (1) for small motions of the particles. Indeed, at the initial instant the particles of the crystal are at the lattice sites \(\mathbf r_i\) with momentum \(\mathbf p_i=0\), and, consequently, their distribution is determined by the function

\[ f_0=\sum_i \delta(\mathbf r-\mathbf r_i)\delta(\mathbf p). \]

With time, the quantities \(\mathbf r_i(t)\), \(\mathbf p_i(t)\), characterizing the state of the \(i\)-th particle, change according to the equations of mechanics, while the particle distribution is determined by the function

\[ f(\mathbf r,\mathbf p,t)=\sum_i \delta(\mathbf r-\mathbf r_i(t))\delta(\mathbf p-\mathbf p_i(t)). \]

For small deviations of the particles from the position of stable equilibrium we have:

\[ m\frac{d\mathbf Q_i}{dt}=\mathbf P_i, \]

\[ \frac{d\mathbf P_i}{dt} = \nabla_{\mathbf r_i}\sum_j \left( \nabla_{\mathbf r_i}\Phi(|\mathbf r_i-\mathbf r_j|),\, \mathbf Q_j-\mathbf Q_i \right) - \sum_{\Omega}\nabla_{\mathbf r_i}V_{\Omega}(\mathbf r_i)e^{i\Omega t+\varepsilon t}, \]

\[ f(\mathbf r,\mathbf p,t) = \sum_i \delta(\mathbf r-\mathbf r_i)\delta(\mathbf p) - \sum_i \left( \nabla_{\mathbf r}\delta(\mathbf r-\mathbf r_i),\,\mathbf Q_i \right)\delta(\mathbf p) - \]

\[ - \sum_i \delta(\mathbf r-\mathbf r_i) \left( \nabla_{\mathbf p}\delta(\mathbf p),\,\mathbf P_i \right) = f_0+\varphi(\mathbf r,\mathbf p,t), \]

where the variation of the phase density determined by equation (1) is equal to (2). Hence it is clear that the linear spectrum of natural oscillations of the system of particles is the Born spectrum. Moreover, since the solution of the mechanical problem is unique, the particle distribution for small oscillations and the found \(\varphi(\mathbf r,\mathbf p,t)\) are unique.

Thus, the linearized Vlasov equation for the case of a crystal at zero temperature has only the Born oscillation spectrum. One can also arrive at this conclusion by solving equation (1) directly. The same result also holds in the quantum-mechanical approach: starting from the varied equations of the self-consistent field, one can obtain a spectrum of oscillations which, in the limit \((\hbar\to 0)\), passes into the Born spectrum. It is interesting to note that, in contrast to the linearized equation, the nonlinear Vlasov equation has a very extensive class of solutions of the most varied type.

I express my gratitude to Academician N. N. Bogolyubov for a valuable discussion of the present note.

Moscow State University
named after M. V. Lomonosov

Received
11 IX 1962

CITED LITERATURE

\(^{1}\) A. Vlasov, J. Phys., 9, 130 (1945); A. A. Vlasov, Theory of Many Particles, 1950.
\(^{2}\) S. Tyablikov, ZhETF, 17, 386 (1947).
\(^{3}\) A. A. Vlasov, V. A. Yakovlev, ZhETF, 20, 1109 (1950).
\(^{4}\) M. Born, M. Gepppert-Mayer, Theory of Solids, 1938.
\(^{5}\) S. V. Tyablikov. Dissertation, Moscow State University, 1947.