DOI: https://doi.org/10.30970/jps.01.25
STATISTICAL OPERATOR OF A SYSTEM OF IDENTICAL INTERACTING PARTICLES IN COORDINATE REPRESENTATION
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Journal of Physical Studies 1(1), 25–38 (1996)
DOI: https://doi.org/10.30970/jps.01.25 STATISTICAL OPERATOR OF A SYSTEM OF IDENTICAL INTERACTING PARTICLES IN COORDINATE REPRESENTATION |
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I. O. Vakarchuk
Ivan Franko Lviv State University, Chair of Theoretical Physics
12 Drahomanov Str., Lviv UA-290005, Ukraine
A new representation of the total density matrix of N identical interacting particles in coordinate representation was derived. It permits to obtain a symmetric matrix in calculations within the approximate aproaches. For a standard representation that follows from the definition the problem of non-symmetry arises due to the use of the approximate methods. To restore the symmetry it is necessary to sum up certain sequences of terms of infinite series of the perturbation theory.
On the basis of a new representation the density matrix in coordinate-momentum representation was calculated. For the logarithm of the kernel of the density matrix integral representation a functional series with the coefficient functions calculated explicitly in the approximation of pair interparticle correlations was obtained. As a result the density matrix integrated with respect to momenta was presented as a product of the density matrix of ideal Bose gas R_N^0 (x ' | x) with the particles changed due to the interaction mass and the P (x ' | x) factor taking into account the interaction of atoms at small distances. At low temperatures the latter is a product of the ground state wave functions of the many-boson system $ψ_0 (x)$, $x = ({\bf r}_1, \ldots , {\bf r}_N)$ being the particles coordinates: \begin{eqnarray*} && R_N (x | x ' ) = P (x ' | x) R_N^0 (x ' | x), \\ && R_N^0 (x|x ') = \frac{1}{N!} \left[ \frac{m_0^*}{2πβ \hbar^2}\right] ^{3N/2} \sum_Q (\pm )^Q \exp \left[ - \frac{m_0^*} {2β \hbar^2} \sum _{j=1}^N ({\bf r}_j - {\bf r}_{Qj} ')^2 \right] , \\ && \frac{m}{m_0^*} = 1 - \frac{1}{3 N} \sum_{{\bf q}\ne 0} \frac{(S_q - 1)^2}{S_q + 1}, \end{eqnarray*} where $S_q$ is the structure factor of the liquid; the summation over $Q$ is done with respect to all the permutations of different indeces $j$; the upper (lower) sign corresponds to Bose- (Fermi-) statistics.
The explicit form of the ground state wave functions and the calculation of the effective mass for liquid $^{4}He$ as well as for the two model systems are given in the present paper. The calculation of the density matrix for the many-boson system was performed within the so-called condensate approximation. Within the framework of this approximation only one state with zero momenta of the particles of a complete set of the basic wave function of a system of free particles is taken into consideration. It appears that this approximation yields correct low-temperature behaviour and a classical expression for the thermodynamical functions.
The explicit dependence of the effective mass on temperature, local density, statistics of particles as well as contributions that account explicitly for the exchange-correlation effects was found. An increase in the particle mass due to renormalization resulted in the decrease in Bose-condensation temperature in many-boson systems (in particular for liquid $^{4}$He, $m_0^{*}\simeq 1.7m$). Taking into account the non-penetrability of atoms leads to the decreasing of free volume and then to a rise in Bose-condensation temperature. These two competitive mechanisms permit to obtain the observable dependence of transition temperature on density in liquid $^{4}$He. Within the approach suggested a possibility of changing the Gaussian type free-particle kernel due to the accounting for many-particle correlations arises. This may lead to changes of the temperature dependence of the thermodynamical functions in the vicinity transition for liquid $^{4}$He.