Finite Size Effects in Bose-Einstein Systems and the Properties of Thin Helium Films.

The distinction is made between a condensation temperature in the thermodynamic sense, where the thermodynamic functions have a discontinuity, and an abrupt accumulation (in the statistical mechanical sense) of particles in the ground state of a system. The criterion for the latter is that an integral will not approximate one term of a sum. The problem of a thin film of gas is solved to illustrate this distinction. For films thicker than a critical thickness a condensation occurs at the same temperature as that for bulk material. The thinner films are effectively two dimensional and have an abrupt accumulation temperature lower than the bulk condensation temperature. Films of liquid helium are briefly considered.The distinction is made between a condensation temperature in the thermodynamic sense, where the thermodynamic functions have a discontinuity, and an abrupt accumulation (in the statistical mechanical sense) of particles in the ground state of a system. The criterion for the latter is that an integral will not approximate one term of a sum. The problem of a thin film of gas is solved to illustrate this distinction. For films thicker than a critical thickness a condensation occurs at the same temperature as that for bulk material. The thinner films are effectively two dimensional and have an abrupt accumulation temperature lower than the bulk condensation temperature. Films of liquid helium are briefly considered.

Physical ReviewAug 1, 194955 citations

Finite size effects in Bose-Einstein assemblies

Raj K. Pathria

Physics Letters AJun 1, 197113 citations

Bose-Einstein condensation in a two-dimensional system at constant pressure

A. N. Chaba, Raj K. Pathria

Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\ensuremath{\simeq}{T}_{c}$, is approached from above, the volume of the system becomes subextensive, ${V}_{c}=O(\frac{N}{\mathrm{ln}N})$ , which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume $V(={V}_{c})$ , the condensate fraction gradually builds up and becomes nonnegligible as $T$ approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below ${T}_{0}$ , it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant $P$ its volume, over a thermodynamically negligible range of temperatures, would reduce to values $O({N}^{\frac{1}{2}})$ which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat ${C}_{P}$ , after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\ensuremath{\simeq}{T}_{c}$, is approached from above, the volume of the system becomes subextensive, ${V}_{c}=O(\frac{N}{\mathrm{ln}N})$, which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume $V(={V}_{c})$, the condensate fraction gradually builds up and becomes nonnegligible as $T$ approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below ${T}_{0}$, it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant $P$ its volume, over a thermodynamically negligible range of temperatures, would reduce to values $O({N}^{\frac{1}{2}})$ which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat ${C}_{P}$, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.

Physical review. B, Solid stateNov 1, 197514 citations

Finite-Size Effects in Bose-Einstein Assemblies.

Stanley Greenspoon

Jan 1, 19750 citations

Bose-Einstein condensation in finite noninteracting systems: A relativistic gas with pair production

Surjit Singh, Raj K. Pathria

Taking into account the possibility of particle-antiparticle pair production, we have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to restricted geometries. Through an extensive use of the Poisson summation formula, we have carried out an explicit evaluation of the summations over states appearing in the problem, which enables us to make a rigorous analysis of the temperature dependence of the thermogeometric parameter $y$ of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\ensuremath{\rho}}_{0}}{\ensuremath{\rho}}$ as a smooth function of temperature from $T\ensuremath{\gtrsim}{T}_{c}$ down to $T=0$ K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.Taking into account the possibility of particle-antiparticle pair production, we have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to restricted geometries. Through an extensive use of the Poisson summation formula, we have carried out an explicit evaluation of the summations over states appearing in the problem, which enables us to make a rigorous analysis of the temperature dependence of the thermogeometric parameter $y$ of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\ensuremath{\rho}}_{0}}{\ensuremath{\rho}}$ as a smooth function of temperature from $T\ensuremath{\gtrsim}{T}_{c}$ down to $T=0$ K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.

Physical Review AJul 1, 198426 citations

THEORY OF A NONIDEAL BOSE GAS AND THE SIZE EFFECTS IN SUPERFLUID FILMS.

Edouard B Sonin

Jan 1, 19690 citations

{V}_{c}=O(\frac{N}{\mathrm{ln}N})

, which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume

V(={V}_{c})

, the condensate fraction gradually builds up and becomes nonnegligible as

T

approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below

{T}_{0}

, it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant

P

its volume, over a thermodynamically negligible range of temperatures, would reduce to values

O({N}^{\frac{1}{2}})

which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat

{C}_{P}

, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.

y

of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\ensuremath{\rho}}_{0}}{\ensuremath{\rho}}$ as a smooth function of temperature from $T\ensuremath{\gtrsim}{T}_{c}$ down to

T=0

K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.