The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order ${l}^{\ensuremath{-}2}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\ensuremath{\simeq}10$. In the critical region, the large- $l$ behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order ${l}^{\ensuremath{-}2}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\ensuremath{\simeq}10$. In the critical region, the large-$l$ behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.

Physical Review AAug 1, 197349 citations

Discontinuities of critical amplitude for specific heat

R. Abe, S. Hikami

Physics Letters AAug 1, 19735 citations

Bose-Einstein condensation of a finite number of particles confined to harmonic traps

Raj K. Pathria

The onset of Bose-Einstein condensation in a sample of N atoms confined to harmonic traps is critically examined. The monotonic growth of the condensate fraction f and the nonmonotic variation of the specific heat C of the sample are studied in fine detail. While no temperature is truly critical, our analysis does provide a strong theoretical basis for recognizing a temperature ${T}^{*},$ that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum ${T}_{m}$ (which happens to be another important ``marker'' in this transition), the value of the specific heat at ${T=T}_{m},$ and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].The onset of Bose-Einstein condensation in a sample of N atoms confined to harmonic traps is critically examined. The monotonic growth of the condensate fraction f and the nonmonotic variation of the specific heat C of the sample are studied in fine detail. While no temperature is truly critical, our analysis does provide a strong theoretical basis for recognizing a temperature ${T}^{*},$ that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum ${T}_{m}$ (which happens to be another important ``marker'' in this transition), the value of the specific heat at ${T=T}_{m},$ and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].

Physical Review AAug 1, 199860 citations

Specific Heat Studies of One-Dimensional Systems.

Ta-sheng Wei

Jan 1, 19760 citations

Finite-size effect for the critical point of an anisotropic dimer model of domain walls

Somendra M. Bhattacharjee, John F. Nagle

The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr= ${N}^{2}$ /M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by ${\mathrm{MN}}^{2}$ t/(M+ ${N}^{2}$ ), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=${N}^{2}$/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by ${\mathrm{MN}}^{2}$t/(M+${N}^{2}$), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.

Physical Review AMay 1, 198546 citations

Specific heat of one-dimensional S = 1 system

Tom de Neef

Physics Letters AFeb 1, 197415 citations

l

behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.

{T}^{*},

that has been identified and advocated by previous authors, as the transition temperature of the system. The specific-heat maximum

{T}_{m}

(which happens to be another important ``marker'' in this transition), the value of the specific heat at

{T=T}_{m},

and the ``jump'' in the specific heat of the system when the number N becomes very large are all explicitly evaluated. Wherever comparison is possible, our analytical results are found to be in perfect agreement with the ones obtained numerically by Kirsten and Toms [Phys. Rev. A 54, 4188 (1996)] and by Haugerud, Haugest, and Ravndal [Phys. Lett. A 225, 18 (1997)].

{N}^{2}

/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by

{\mathrm{MN}}^{2}

t/(M+

{N}^{2}

), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.