Ursell Operators in Statistical Physics III: Thermodynamic Properties of Degenerate Gases

Grüter, Peter; Laloë, Franck; Meyerovich, A. E.; Mullin, William J.

doi:10.1051/jp1:1997171

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…and m the mass of the bosons. We will also replace the value of the diagonal matrix elements of U S 2 by their expression as a function of the (symmetrized) Ursell length a S U (k), which is given in equation ( 17) and ( 4) of [43]:…”

Section: A Mean-field Like Approximationmentioning

confidence: 99%

Bose-Einstein condensation in interacting gases

Holzmann¹,

Grüter²,

Laloë³

1999

Eur. Phys. J. B

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104

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We study the occurrence of a Bose-Einstein transition in a dilute gas with repulsive interactions, starting from temperatures above the transition temperature. The formalism, based on the use of Ursell operators, allows us to evaluate the one-particle density operator with more flexibility than in mean-field theories, since it does not necessarily coincide with that of an ideal gas with adjustable parameters (chemical potential, etc.). In a first step, a simple approximation is used (Ursell-Dyson approximation), which allow us to recover results which are similar to those of the usual mean-field theories. In a second step, a more precise treatment of the correlations and velocity dependence of the populations in the system is elaborated. This introduces new physical effects, such as a marked change of the velocity profile just above the transition: low velocities are more populated than in an ideal gas. A consequence of this distortion is an increase of the critical temperature (at constant density) of the Bose gas, in agreement with those of recent path integral Monte-Carlo calculations for hard spheres.

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Section: A Mean-field Like Approximationmentioning

confidence: 99%

Bose-Einstein condensation in interacting gases

Holzmann¹,

Grüter²,

Laloë³

1999

Eur. Phys. J. B

Self Cite

104

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“…The usual way to understand this property is to replace the real potential V 12 by a pseudo-potential, which is directly proportional to the scattering length a and treated to first order, a somewhat heuristic method (since the exact reason why using the real potential is incorrect is not so clear). Here, we clearly see that what appears naturally is the matrix elements of U 2 (1, 2); as shown in [19], the latter can be expressed in terms of phase shifts and be shown to be proportional to the scattering length 10 a: for instance, if the latter is positive, we directly get a positive value for the mean-field, without any special manipulation 11 .…”

Section: Physical Discussionmentioning

confidence: 68%

“…(for the moment, when l > 2, we do not specify the exact relation between an U l operator and U l , but we will come back to this point later). We note that, here, U 2 is defined symmetrically, with square roots of U 1 operators on each side (U 1 is a positive operator), so that U 2 is Hermitian -this was not the case in [19]. The introduction of U 2 has two advantages.…”

Section: Notationmentioning

confidence: 99%

“…First, factorizing the kinetic energy brings the formalism closer to classical statistical mechanics, where kinetic energy always factorizes out exactly; in other words, U 2 (i, j) is the quantum equivalent of f ij , although it is not strictly analogous (in quantum mechanics operators do not necessarily commute, so that the kinetic energy can still play a role in U 2 (i, j)). Second, in the limit of low energies, it is possible to make use of the MIME (momentum independent matrix elements) approximation, where the diagonal matrix elements of U 2 are constants -see for instance the characterization of the diagonal elements of U 2 in terms of the Ursell length in [19]. Graphically, in order to distinguish U 2 's from U 2 's in diagrams, we use two vertical parallel lines in the former case, as in [4] and [5], but only a single line for U 2 's.…”

Section: Notationmentioning

confidence: 99%

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Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

2002

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A widely used formalism in quantum statistical physics is the formalism of Green's functions [1,2,3], where the techniques of second quantization and field theory are used from the beginning; the notion of exchange operators of indistinguishable particles is of course contained, but in a completely implicit way. In the formalism of Ursell operators [4,5,6], the starting point is first quantization with numbered particles, so that the role of exchange cycles becomes completely explicit: these cycles appear clearly in all diagrams and, for instance, they are the only source of diagrams for the ideal quantum gas. This reduces the distance between the formalism and, for instance, numerical calculations such as the PIMC method (Path Integral Monte Carlo), where particles are also numbered and the exchange cycles are explicitly sampled by random choices. Moreover, it becomes possible to assume that the particles obey Boltzmann statistics, just by "switching off the cycles", a task that would be difficult in the Green's function formalism. Needless to say, this does not mean that Green's functions are, in general, less powerful than the Ursell formalism! The opposite is actually closer to reality: for instance, Green's functions handle timedependent problems easily, while this is not the case in the Ursell formalism. But it remains true that, if one is interested in a detailed discussion of the effects of quantum statistics, it becomes more straightforward to resort to the Ursell formalism.In this article, we consider dense systems, for which it is not necessarily possible to limit oneself to first order density effects. In contrast to the situation in a dilute gas, a given particle may interact frequently with several others at the same time, and even liquefaction may take place. Therefore, we will no longer ignore all Ursell operators beyond U 2 , as was done in most of previous work in this formalism; operators U 3 , U 4 , etc. now become important. One may actually wonder what the role of these higher order operators is in general, and why exactly it is possible to ignore their role in a dilute system, as was done in [6] for instance. We will see that part of their contribution (what we will call their tree reducible part) groups naturally with U 2 through a chain of binary interactions, and builds an exponential of the mean-field energy. In other words, instead of making the problem more complicated, this contribution of the higher order operators builds exactly the exponential of U 2 that is needed to reconstruct a simple and natural expression of the mean-field. Nevertheless, this mean-field is expressed in terms of the matrix elements of U 2 , instead of the usual matrix elements of the potential itself; in a sense, what we obtain is the exponential of an exponential, since U 2 itself contains exponentials of the Hamiltonian and corresponds physically to the local change of the Boltzmann equilibrium. As a consequence, the logarithms that appeared in [6], and had to be expanded to first order in density, are actu...

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“…Once this point of view is adopted, the difficulty consists in expressing quantities of physical interest in terms of short-time processes. This will be done using the standard tool of cluster expansions [28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal pair correlations in Lieb-Liniger gases: A unified nonperturbative approach from weak degeneracy to high temperatures

et al. 2018

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We present analytical results for the nonlocal pair correlations in one-dimensional bosonic systems with repulsive contact interactions that are uniformly valid from the classical regime of high temperatures down to weak quantum degeneracy entering the regime of ultralow temperatures. By using the information contained in the short-time approximations of the full many-body propagator, we derive results that are nonperturbative in the interaction parameter while covering a wide range of temperatures and densities. For the case of three particles we give a simple formula for arbitrary couplings that is exact in the dilute limit while remaining valid up to the regime where the thermal de Broglie wavelength λ T is of the order of the characteristic length L of the system. We then show how to use this result to find analytical expressions for the nonlocal correlations for arbitrary but fixed particle numbers N including finite-size corrections. Neglecting the latter in the thermodynamic limit provides an expansion in the quantum degeneracy parameter Nλ T /L. We compare our analytical results with numerical Bethe ansatz calculations, finding excellent agreement.

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Ursell Operators in Statistical Physics III: Thermodynamic Properties of Degenerate Gases

Cited by 6 publications

References 17 publications

Bose-Einstein condensation in interacting gases

Bose-Einstein condensation in interacting gases

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

Nonlocal pair correlations in Lieb-Liniger gases: A unified nonperturbative approach from weak degeneracy to high temperatures

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