The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of 'hotness' is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law.
The ground state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here the first proof of the asymptotic exactness of this approximation for the ground state energy and particle density of a dilute Bose gas with a positive interaction.
The ground state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here the first proof of the asymptotic exactness of this approximation for the ground state energy and particle density of a dilute Bose gas with a positive interaction.
We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representation and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.
Now that the properties of low temperature Bose gases at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. One of these is that the leading term in the energy/particle is 2πh 2 ρa/m, where a is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.PACS numbers: 05.30. Jp, 03.75.Fi, With the renewed experimental interest in low density, low temperature Bose gases, some of the formulas posited four and five decades ago have been dusted off and reexamined. One of these is the leading term in the ground state energy. In the limit of small particle density, ρ,where e 0 (ρ) is the ground state energy (g.s.e.) per particle in the thermodynamic limit, a is the scattering length (assumed positive) of the two-body potential v for bosons of mass m, and µ ≡h 2 /2m. Is (1) correct? In particular, is it true for the hardsphere gas? While there have been many attempts at a rigorous proof of (1) in the past forty years, none has been found so far. Our aim here is to supply that proof for finite range, positive potentials. As remarked below, (1) cannot hold unrestrictedly; more than a > 0 is needed.An upper bound for e 0 (ρ) agreeing with (1) is not easy to derive, but it was achieved for hard spheres by a variational calculation [1], which can be extended to include general, positive potentials of finite range. What remained unknown was a good lower bound. The only one available is Dyson's [1], and that is about fourteen times smaller than (1). In this paper we shall provide a lower bound of the desired form, and thus prove (1). We can also give explicit error bounds for small enough values of the dimensionless parameter Y ≡ 4πρa 3 /3:for some fixed C (which is not evaluated explicitly because C and the exponent 1/17 are only of academic interest). The bound (2) holds for all non-negative, finite range, spherical, two-body potentials. A typical experimental value [2] is Y ≈ 10 −5 . Dyson's upper bound is µ 4πρa(1 + 2Y 1/3 )(1 − Y 1/3 ) −2 . We conjecture that (1) requires only a positive scattering length and the absence of any many-body, negative energy bound state. If there are such bound states then (1) is certainly wrong, but this obvious caveat does not seem to have been clearly emphasized before. There is a 'nice' potential with positive scattering length, no 2-body bound state, but with a 3-body bound state [3].Our method also obviously applies to the positive temperature free energy (because Neumann boundary conditions give an upper bound to the solution to the heat (or Bloch) equation).We also give some explicit bounds for finite systems, which might be useful for experiments with traps, but we concentrate here on the thermodynamic limit for simplicity. For traps with slowly varying...
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