Theory of superfluidity

D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty.

The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.

Related papers

Rafael F. Angulo G.

A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained. D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty. The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained. D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty. The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.

Sep 15, 20260 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an eﬀective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in diﬀerent, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$ -norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$ Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an eﬀective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in diﬀerent, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$

Dec 9, 20250 citations

A new view on superfluidity

В. А. Головко

This paper represents the full version of a paper published earlier in Physica A [246 (1997), 275]. The present paper includes argumentation, proofs and details omitted in the shortened version. The papers are a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose-Einstein condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the papers superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The equations obtained are solved in two simple cases. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well.This paper represents the full version of a paper published earlier in Physica A [246 (1997), 275]. The present paper includes argumentation, proofs and details omitted in the shortened version. The papers are a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose-Einstein condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the papers superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The equations obtained are solved in two simple cases. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well.

arXiv:0902.4134Feb 24, 20090 citations

Condensation ofNbosons. II. Nonequilibrium analysis of an ideal Bose gas and the laser phase-transition analogy

Vitaly V. Kocharovsky, Marlan O. Scully, Shi-Yao Zhu +1 more

A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.

Physical Review AJan 20, 200064 citations

The Superfluid State of a Bose Liquid as a Superposition of a Single-Particle and Pair Coherent Condensates

Ernst A. Pashitskii, Stefan V. Mashkevich, S. I. Vilchynskyy

One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid $n_0/n\ll 1$ that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, $(n-n_0)/n_0\ll 1$ . A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid $^4$ He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid $n_0/n\ll 1$ that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, $(n-n_0)/n_0\ll 1$. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid $^4$He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.

arXiv:cond-mat/0104415Apr 22, 20010 citations

Conclusion

Philippe Nozières, David Pines

We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.

CRC Press eBooksMar 5, 20180 citations

L^2

-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others.

\textit{Full-text embargoed until: 2026-01-01}

n_0/n\ll 1

that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC,

(n-n_0)/n_0\ll 1

. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid

^4

He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.