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Physical Review C

Variational description of the nuclear free energy

D.H. Feng, Robert Gilmore, Lorenzo M. Narducci

Published Mar 1, 1979• 32 citations

Abstract

By means of a variational calculation, we place an upper bound on the finite-temperature free energy for nuclear systems which can be described by pseudospin Hamiltonians. The trial states are irreducible permutation invariant Gibbs states. The best trial state is the one which minimizes the free energy operator. We compare the upper bound with the numerically computed free energy for the Meshkov-Glick-Lipkin Hamiltonian for various values of nucleon number $N$ and nuclear interaction strength $V$. For large $N$ and/or $\ensuremath{\beta}(=\frac{1}{\mathrm{kT}})$ the best trial Gibbs state becomes a good approximation to the actual density operator. Somewhat surprisingly, the variational approach reveals the presence of a second order thermodynamic phase transition much more clearly than the numerical computation does, even though the former is only an approximation to the latter.NUCLEAR STRUCTURE Finite-temperature free energy, pseudospin Hamiltonian, variational description and phase transitions, atomic coherent states.

DOI: 10.1103/physrevc.19.1119

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Physical Review C•Aug 1, 1984•75 citations

By means of a variational calculation, we place an upper bound on the finite-temperature free energy for nuclear systems which can be described by pseudospin Hamiltonians. The trial states are irreducible permutation invariant Gibbs states. The best trial state is the one which minimizes the free energy operator. We compare the upper bound with the numerically computed free energy for the Meshkov-Glick-Lipkin Hamiltonian for various values of nucleon number

N

and nuclear interaction strength

V

. For large

N

and/or

\ensuremath{\beta}(=\frac{1}{\mathrm{kT}})

the best trial Gibbs state becomes a good approximation to the actual density operator. Somewhat surprisingly, the variational approach reveals the presence of a second order thermodynamic phase transition much more clearly than the numerical computation does, even though the former is only an approximation to the latter.NUCLEAR STRUCTURE Finite-temperature free energy, pseudospin Hamiltonian, variational description and phase transitions, atomic coherent states.

The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.

A trial wavefunction is constructed for nuclear matter in terms of the reaction matrix. The trial function includes only two-particle excitations and the resulting energy expectation value for two-nucleon forces with a state-independent repulsive core of Yukawa shape approaches in the thermodynamic limit the energy given by the lowest-order Brueckner theory with standard dispersion. Evaluation of the energy expectation value is carried out in a systematic way by use of an identity that holds between the Hamiltonian and reaction matrix.

The Lieb-Mattis theorem orders the lowest-energy states of total spin s of a system of P interacting fermions. We generalize these predictions to fermionic mixtures of P particles with more than N=2 spinor components or species in the thermodynamic limit P→∞. The lowest-energy state inside each permutation symmetry sector h, arising in the P-fold tensor product decomposition, is well approximated by a U(N) coherent (quasiclassical variational) state, especially in the limit P→∞. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau h_{0}) configuration. We exemplify our construction with the N=3 level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and U(N)-invariant quantum Hall ferromagnets. In the limit P→∞, each lowest-energy state within each permutation symmetry sector h undergoes a quantum phase transition for a critical value λ_{c}(h) of the exchange coupling constant λ, depending on h. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector h_{0}.

Methods which include the effect of mean-field fluctuations on the nuclear partition function and level density of hot nuclei are tested by applying them to a many-body system for which exact solutions are available, the Lipkin model. It is found that for finite systems fluctuation corrections to the mean field partition function and level density can be substantial, and especially so near a shape phase transition where these fluctuations smooth out discontinuities in the mean-field quantities. The model calculations suggest that only time-independent fluctuations are important at high temperatures or at high excitation energies. Moreover, even when fluctuations are restricted to a small set of constraints, a significant improvement over the mean-field calculations can result. This is thus a practical approximation since it requires only knowledge of the deformation free energy surfaces at finite temperature.