We analyze the nonequilibrium dynamics of the $O(N) {\ensuremath{\Phi}}^{4}$ model in the large N limit with a broken symmetry tree level potential and for states of large energy density. The dynamics is dramatically different when the energy density is above the top of the tree level potential ${V}_{0}$ than when it is below it. When the energy density is below ${V}_{0},$ we find that nonperturbative particle production through spinodal instabilities provides a dynamical mechanism for the Maxwell construction. The asymptotic values of the order parameter only depend on the initial energy density and all values between the minima of the tree level potential are available; the asymptotic dynamical ``effective potential'' is flat between the minima. When the energy density is larger than ${V}_{0},$ the evolution samples ergodically the broken symmetry states, as a consequence of nonperturbative particle production via parametric amplification. Furthermore, we examine the quantum dynamics of phase ordering into the broken symmetry phase and find a novel scaling behavior of the correlation function. There is a crossover in the dynamical correlation length at a time scale ${t}_{s}\ensuremath{\approx}\mathrm{ln}(1/\ensuremath{\lambda}).$ For $t<{t}_{s}$ the dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}\sqrt{t}$ and the evolution is dominated by linear instabilities and spinodal decomposition, whereas for $t>{t}_{s}$ the evolution is nonlinear and dominated by the onset of nonequilibrium Bose-Einstein condensation of long-wavelength Goldstone bosons. In this regime a true scaling solution emerges with a nonperturbative anomalous scaling length dimension $z=1/2$ and a dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}(t\ensuremath{-}{t}_{s}).$ The equal time correlation function in this scaling regime vanishes for $r>2(t\ensuremath{-}{t}_{s})$ by causality. For $t>{t}_{s}$ phase ordering proceeds by the formation of domains that grow at the speed of light, with nonperturbative condensates of Goldstone bosons and the equal time correlation function falls of as $1/r.$ A semiclassical but stochastic description emerges for time scales $t>{t}_{s}.$ Our results are compared to phase ordering in classical stochastic descriptions in condensed matter and cosmology.We analyze the nonequilibrium dynamics of the $O(N) {\ensuremath{\Phi}}^{4}$ model in the large N limit with a broken symmetry tree level potential and for states of large energy density. The dynamics is dramatically different when the energy density is above the top of the tree level potential ${V}_{0}$ than when it is below it. When the energy density is below ${V}_{0},$ we find that nonperturbative particle production through spinodal instabilities provides a dynamical mechanism for the Maxwell construction. The asymptotic values of the order parameter only depend on the initial energy density and all values between the minima of the tree level potential are available; the asymptotic dynamical ``effective potential'' is flat between the minima. When the energy density is larger than ${V}_{0},$ the evolution samples ergodically the broken symmetry states, as a consequence of nonperturbative particle production via parametric amplification. Furthermore, we examine the quantum dynamics of phase ordering into the broken symmetry phase and find a novel scaling behavior of the correlation function. There is a crossover in the dynamical correlation length at a time scale ${t}_{s}\ensuremath{\approx}\mathrm{ln}(1/\ensuremath{\lambda}).$ For $t<{t}_{s}$ the dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}\sqrt{t}$ and the evolution is dominated by linear instabilities and spinodal decomposition, whereas for $t>{t}_{s}$ the evolution is nonlinear and dominated by the onset of nonequilibrium Bose-Einstein condensation of long-wavelength Goldstone bosons. In this regime a true scaling solution emerges with a nonperturbative anomalous scaling length dimension $z=1/2$ and a dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}(t\ensuremath{-}{t}_{s}).$ The equal time correlation function in this scaling regime vanishes for $r>2(t\ensuremath{-}{t}_{s})$ by causality. For $t>{t}_{s}$ phase ordering proceeds by the formation of domains that grow at the speed of light, with nonperturbative condensates of Goldstone bosons and the equal time correlation function falls of as $1/r.$ A semiclassical but stochastic description emerges for time scales $t>{t}_{s}.$ Our results are compared to phase ordering in classical stochastic descriptions in condensed matter and cosmology.

arXiv:hep-ph/9811273Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMay 13, 199963 citations

Energy fluctuation and discontinuity of specific heat

Shyamal Biswas, Joydip Mitra, Saugata Bhattacharyya

The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.

arXiv:1502.05542Journal of Statistical Mechanics Theory and ExperimentMar 11, 20154 citations

The Quantum-Classical Crossover as Phase Transition

WORLD SCIENTIFIC eBooksMar 1, 20060 citations

Exact boson representation of quantum spin systems and investigation of their critical behavior

Isaac Goldhirsch, Evgeny Levich, Victor Yakhot

Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at ${T}_{c}$ the quantum $S=\frac{1}{2}$ spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at ${T}_{c}$ the quantum $S=\frac{1}{2}$ spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.

Physical review. B, Condensed matterMay 1, 197933 citations

Theory of Quantum-Classical Crossover

Artur Mendes Kur

Jan 1, 20260 citations

{V}_{0}

than when it is below it. When the energy density is below

{V}_{0},

we find that nonperturbative particle production through spinodal instabilities provides a dynamical mechanism for the Maxwell construction. The asymptotic values of the order parameter only depend on the initial energy density and all values between the minima of the tree level potential are available; the asymptotic dynamical ``effective potential'' is flat between the minima. When the energy density is larger than

{V}_{0},

the evolution samples ergodically the broken symmetry states, as a consequence of nonperturbative particle production via parametric amplification. Furthermore, we examine the quantum dynamics of phase ordering into the broken symmetry phase and find a novel scaling behavior of the correlation function. There is a crossover in the dynamical correlation length at a time scale ${t}_{s}\ensuremath{\approx}\mathrm{ln}(1/\ensuremath{\lambda}).$ For

t<{t}_{s}

the dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}\sqrt{t}$ and the evolution is dominated by linear instabilities and spinodal decomposition, whereas for

t>{t}_{s}

the evolution is nonlinear and dominated by the onset of nonequilibrium Bose-Einstein condensation of long-wavelength Goldstone bosons. In this regime a true scaling solution emerges with a nonperturbative anomalous scaling length dimension

z=1/2

and a dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}(t\ensuremath{-}{t}_{s}).$ The equal time correlation function in this scaling regime vanishes for $r>2(t\ensuremath{-}{t}_{s})$ by causality. For

t>{t}_{s}

phase ordering proceeds by the formation of domains that grow at the speed of light, with nonperturbative condensates of Goldstone bosons and the equal time correlation function falls of as

1/r.

A semiclassical but stochastic description emerges for time scales

t>{t}_{s}.

Our results are compared to phase ordering in classical stochastic descriptions in condensed matter and cosmology.

{T}_{c}

the quantum

S=\frac{1}{2}

spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.