数理解析研究所講究録 1227 巻 2001 年 1-13

1Superfluidity in the stochastic limit
L. Accardi, S.V. Kozyrev
May 10, 2001
Centro Vito Volterra, Universita di Roma Tor Vergata
Abstract
In the present paper we outline the stochastic limit approach to superfluidity. The
Hamiltonian describing the interaction between the Bose condensate and the normal phaseis introduced. Sufficient in the stochastic limit condition of superfluidity is proposed.
Existence of superfluidity in the stochastic limit of this system is proved and the nonlinear (quadratic) equation of motion describing the superfluid liquid is obtained.

1Introduction
The theory of superfluidity was developed by L.D.Landau and N.N.Bogoliubov, cf. [1], [2],[3], [4], for an introduction see [5]. An analogue of the approach of [3], [4] was applied tosuperconductivity theory, cf. [6].
The essence of the superfluidity phenomenon is that the Bose condensate becomes superfluidand friction between condensate and normal phase disappears. N.N.Bogoliubov in [3], [4] foundthat this can be explained as an effect of stabilization of the condensate by interaction between
particles.
In the present paper, using as astarting point the approach of [3], [4], we introduce an
Hamiltonian that describes the interaction between the Bose condensate and the normal phaseand investigate this Hamiltonian using the stochastic limit approach cf. [7].
The name stochastic limit is due to the property that in this approach the quantum field
is approximated by aquantum white noise and the Schr\"odinger equation is approximated by awhite noise Hamiltonian equation. In the stochastic limit we start from an Hamiltonian of theform$H=H_{0}+\lambda H_{I}$in the solution of thewhere Ais acoupling constant and we make the time rescaling, associated to the Hamiltonian
Schr\"odinger equation in interaction picture$tarrow t/\lambda^{2}$$U_{t}^{(\lambda)}=e^{itH_{0}}e^{-itH}$$H$,$\mathrm{i}.\mathrm{e}$.
$\frac{\partial}{\partial t}U_{t}^{(\lambda)}=-i\lambda H_{I}(t)U_{t}^{(\lambda)}$,$U_{0}^{(\lambda)}=1$

2with $H_{I}(t)=e^{itH_{0}}H_{I}e^{-itH_{0}}$ (the evolved interaction Hamiltonian). This leads to the rescaled
equation$\frac{\partial}{\partial t}U_{t/\lambda^{2}}^{(\lambda)}=-\frac{i}{\lambda}H_{I}(t/\lambda^{2})U_{t/\lambda^{2}}^{(\lambda)}$and one wants to study the limits, in atopology to be specified,$\lim_{\lambdaarrow 0}H_{I}(t/\lambda^{2})=H_{t}$$\lim_{\lambdaarrow 0}U_{t/\lambda^{2}}^{(\lambda)}=U_{t}$Moreover one wants to prove that

$U_{t}$

is the solution of the white noise Hamiltonian equation$\partial_{t}U_{t}=-iH_{t}U_{t}$;$U_{0}=1$

which is equvalent to aquantum stochastic differential equation.
The structure of the present paper is as follows. In section 2we discuss the Bogoliubov
Landau condition of superfluidity. In sections 3and 4we review some basic facts from [5] onBose condensation in ideal and non-ideal Bose gases. In section 5we propose the Hamiltonianof condensate-normal phase interaction and investigate this model using the stochastic limit.
In section 6we construct by the stochastic golden rule the master equation for the condensate
normal phase interaction and prove the existence of superfluidity in this system. We prove thatthe standard Bogoliubov-Landau condition is not sufficient to garantee the superfluidity in thestochastic limit and introduce aprocedure to overcome this problem. In section 7we sum up
the main conclusion that can be draw from our results.

2The Landau and Bogoliubov ideas on superfluidity
theoryTissa and London conjectured that the existence of the Bose condensate can explain superfluidity in the sense that the condensate corresponds to the superfluid component and thetemperature phase corresponds to normal component.
By Landau’s argument [1], [2] the existence of acondensate itself is not enough to provesuperfluidity and moreover there is no superfluidity in ideal Bose gas.
Consider the condensate moving with velocity and the non-moving normal state. Supposethat the friction of the condensate with the normal phase stimulates the transition of aparticle. Then thisin condensate from the state with velocity to the state with velocity. Since the. Before the transition the particle had energy
particle will have energy
particle
theoftransition is due to friction with normal state this corresponds to the excitation. We get the energy differenceof the normal phase with energy$u$$u-\mathrm{A}m$$u$$\frac{mu^{2}}{2}$$\frac{(mu-p)^{2}}{2m}$$L^{2}2\overline{m}$Energy after$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}+\mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}$of excitation –Energy before transition$= \frac{(mu-p)^{2}}{2m}+\frac{p^{2}}{2m}-\frac{mu^{2}}{2}=\frac{p^{2}}{m}$$- pu<0$for small

$p$

$=$

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This means that such transitions are energetically possible and that the condensate is nonstable.
Bogoliubov found [3], [4] that the interaction between the particles in thecondensate may
stabilize the condensate and modify the energy difference in such away that transitions toslower motion due to friction will proceed with positive energy difference and therefore will beenergetically forbidden.
Bogoliubov found that interaction makes the state of normal phase instable. To stabilize
the state one has to make aBogoliubov transformation. This procedure changes the dispersion$E(k)$ of excitations of normal phase, cf. [3], [4], [5], [6].
After the canonical transformation the energy difference for transitionis givenby$E(k)+\epsilon(p-k)-\epsilon(p)$(1)$p\vdash\Rightarrow p-k$

and $E(p)$ has different dependence on . In this case one can overcome thewhere
Landau objection. Consider the following examples.
$\epsilon \mathrm{i}(p)$$= \frac{p^{2}}{2m}$$p$EXAMPLE 1: Consider radiative dispersion $E(k)=c|k|+\ldots$ .In this case $E(k)$ for smalland (1) takes the form
enough is proportional to
$k$

$|k|$

$c|k|-pk+ \frac{1}{2}k^{2}>0$

EXAMPLE 2:,$|p|<c$(2)

Consider the polaron model with $E(k)=\omega$ . In this case (1) becomes$\omega$$-kp+ \frac{1}{2}k^{2}>0$,$|p|<\sqrt{\omega}$

Let us show that the results of Landau and Bogoliubov discussed above are connected withthe stochastic limit approach. One of important properties of the stochastic limit is that themain properties of the dynamics is controlled by -functions of energy differences of the form$\delta$$\delta(E(k)+\epsilon(p-k)-\epsilon(p))$(3)(this coincides with the of (1)). The possibility to control the dynamics by exploiting such-functions was called in [8] the Cheshire Cat effect. We find that the Bogoliubov-Landau
condition of superfluidity is connected with the Cheshire Cat effect developed in the stochastic
limit approach.
$\delta$$\delta$

3Condensation of an ideal Bose gas
In the present section we discuss standard material on Bose condensation of an ideal Bose gas,cf. [5]. The Hamiltonian of an ideal Bose gas in second quantization is given by$H= \int\omega(p)b^{*}(p)b(p)dp$

The grand canonical ensemble Hamiltonian is$\Gamma=H-\mu N$ ,$N=[$ $b^{*}(p)b(p)dp$

4
The equilibrium state of the gas is the unique mean zero gauge invariant state with expectations$\langle b^{*}(p)b(p)\rangle=\langle n(p)\rangle=\frac{1}{e^{\beta(\omega(p)-\mu)}-1}$

giving the density of particles with momentum and whereis the inverse temperature.
We takesince.
The integral over the density of degrees of freedom of gas is equal to the density of the gas:$p$$\mu\leq 0$$\beta=\frac{1}{\theta}$$\langle n(p)\rangle\geq 0$$\int\langle n(p)\rangle dp=\rho$

To describe Bose condensation, we consider the zero mode of the field
the other modes$b_{0}$

as singled out from$[b_{0}, b_{0}^{*}]=1$

and take, as the reference state of the field, the state not the equilibrium one but agaussianstate of the form$\langle n(p)\rangle=c\delta(p)+\frac{1}{e^{\beta\omega(p)}-1}$

Notice that the chemical potential in this formula is taken equal to zero.
We consider the constant $c=c(\theta)$ depending on the temperature. To determine thisdependence we use the conservation of number of particles, given by the following:(4)$c+ \int\frac{1}{e^{\beta\omega(p)}-1}dp=\rho$Therefore$c= \rho(1-(\frac{\theta}{\theta_{c}})^{\frac{3}{2}})$. The critical temperature is the temperature when thewhere $d=3$ andintegral in (4) becomes equal to the total density of particles and condensate disappears.
This implies the state of the Bose gas in the form$\omega(p)=R_{\frac{2}{m}}2’\theta\leq\theta_{c}$$\theta_{c}$$\rho$$\langle n(p)\rangle=\rho(1-(\frac{\theta}{\theta_{c}})^{\frac{3}{2}})\delta(p)+\frac{1}{e^{\beta\omega(p)}-1}$

4Condensation of anon-ideal Bose gas
In the present section we review material from [5] containing the discussion of the Bose condensection of non-ideal Bose gas and prove the applicability of the Bogoliubov-Landau conditionof stability of the condensate.
The second quantized Hamiltonian of anon-ideal Bose gas (for the grand canonical ensemble) is:$\Gamma=\sum_{p}(\omega(p)-\mu)b^{*}(p)b(p)+\frac{\lambda}{2V}\sum_{p_{1}+p_{2}=p_{1}’+p_{\acute{2}}}g(p_{1}-p_{1}’)b^{*}(p_{1})b^{*}(p_{2})b(p_{2}’)b(p_{1}’)$

5
We consider astate such that almost all particles with $p=0$ are in the condensate. This implies$\langle b_{0}^{*}b_{0}\rangle=N_{0}$where we denote$b_{0}^{*}$,$N_{0}>>1$

the creation of the particle with $p=0$ . Since$\langle b_{0}b_{0}^{*}\rangle=N_{0}+1$

and $N_{0}+1$ is almost equal to No, we can consider$|b_{0}|^{2}=b_{0}^{*}b_{0}=b_{0}^{*}b_{0}=N_{0}$$b_{0}$

as aclassical (commuting) variable with,$b_{0}=\sqrt{N_{0}}e^{i\phi}$This allows to simplify the Hamiltonian . Since $b(p)$ with $p\neq 0$ are not in the condensate,
amplitudes of $b(p)$ are small with respect to . Therefore we can keep in the Hamiltonan
only the terms of second order in $b(p)$ , $p\neq 0$ and skip higher order terms (3 and 4).
This mean field approximation will be valid only when almost all particles are in the condensate, i.e. for atemperature close to zero: $\theta=0$ .
In this approximation the fourth order term in takes the form$\Gamma$$\Gamma$$b_{0}$$\Gamma$$\frac{\lambda}{2V}(g(0)N_{0}^{2}+b_{0}^{*2}\sum_{p}g(p)b(p)b(-p)+b_{0}^{*}b_{0}\sum_{p}(g(p)+g(-p))b(p)b(p)+$$+2b_{0}^{*}b_{0} \sum_{p}g(0)b(p)b(p)+b_{0}^{2}\sum_{p}g(p)b^{*}(p)b^{*}(-p))$Using the conditions $g(p)=g(-p)$ , $\omega(0)=0$ we get for$\Gamma$$\Gamma=\frac{\lambda N_{0}^{2}}{2V}g(0)-\mu N_{0}+\sum_{p\neq 0}(\omega(p)-\mu+\frac{\lambda N_{0}}{V}g(0)+\frac{\lambda N_{0}}{V}g(p))b^{*}(p)b(p)+$$+ \frac{\lambda}{2V}b_{0}^{*2}\sum_{p}g(p)b(p)b(-p)+\frac{\lambda}{2V}b_{0}^{2}\sum_{p}g(p)b^{*}(p)b^{*}(-p)$

This implies that the ground state of$\Gamma$

has the energy$E_{1}= \frac{\lambda N_{0}^{2}}{2V}g(0)-\mu N_{0}$The chemical potential is defined by the equation$\frac{\partial E_{1}}{\partial N_{0}}=0$

and in our case it is equal to$\mu=\frac{\lambda N_{0}}{V}g(0)$$b(p)-+b(p)e^{i\phi}$ we cancel
With this chemical potential and making the canonical transformation
so that the Hamiltonian becomesthe phase of the complex number$b_{0}=\sqrt{N_{0}}e^{i\phi}$$\Gamma$$\Gamma=-\frac{\lambda_{\mathit{1}}\mathrm{V}_{0}^{2}}{2V}g(0)+\sum_{p\neq 0}(\omega(p)+\frac{\lambda N_{0}}{V}g(p))b^{*}(p)b(p)+\frac{\lambda N_{0}}{2V}\sum_{p\neq 0}g(p)(b^{*}(p)b^{*}(-p)+b(-p)b(p))$

6
Let us make acanonical transformation to diagonalize this quadratic Hamiltonian. We consider
the following transformation$a(p)=u_{p}b(p)+v_{p}b^{*}(-p)$$a^{*}(p)=u_{p}b^{*}(p)+v_{p}b(-p)$$u_{p}^{2}-v_{p}^{2}=1$,,$u_{p}=u_{-p}$ $v_{p}=v_{-p}$;$u_{p}$,$v_{p}\in \mathrm{R}$The inverse transformation is given by$b(p)=u_{p}a(p)-v_{p}a^{*}(-p)$$b^{*}(p)=u_{p}a^{*}(p)-v_{p}a(-p)$

After this canonical transformation the off-diagonal terms in the Hamiltonian (coefficientsofand $a(-p)a(p))$ are equal to$a^{*}(p)a^{*}(-p)$$-u_{p}v_{p}( \omega(p)+\frac{\lambda N_{0}}{V}g(p))+(u_{p}^{2}+v_{p}^{2})\frac{\lambda N_{0}}{2V}g(p)$(5)

and the diagonal terms (coefficients of $a^{*}(p)a(p)$ ) are equal to$(u_{p}^{2}+v_{p}^{2})( \omega(p)+\frac{\lambda N_{0}}{V}g(p))-4u_{p}v_{p}\frac{\lambda N_{0}}{2V}g(p)$Since$u_{p}^{2}-v_{p}^{2}=1$(6)

we can use the hyperbolic parametrization$u_{p}=\mathrm{c}\mathrm{h}x$,$v_{p}=\mathrm{s}\mathrm{h}x$

The compensation equation (vanishing of off-diagonal terms (5)) in this parametrization takes
the formth$2x= \frac{\frac{\lambda N}{V}\alpha_{g(p)}}{\omega(p)+\frac{\lambda N}{V}\not\subset_{g(p)}}$This implies$u_{p}^{2}+v_{p}^{2}= \mathrm{c}\mathrm{h}2x=\frac{1}{\sqrt{1-(\frac{\underline{\lambda}N_{\Delta_{g(p)}}V}{\omega(p)+_{V}^{\underline{\lambda}N_{\Delta}}g(p)})^{2}}}$$2u_{p}v_{p}= \mathrm{s}\mathrm{h}2x=\frac{1}{\sqrt{1-(\frac{\frac{\lambda N}{v}\mathrm{n}_{g(p)}}{\omega(p)+_{V}^{\underline{\lambda}N\mathrm{g}}g(p)})^{2}}}\frac{\underline{\lambda}NV\mathrm{I}g(p)}{\omega(p)+\mathrm{I}gV(\underline{\lambda}Np)}$

Therefore for (6) we get$E(p)=\sqrt{\omega^{2}(p)+\frac{2\lambda N_{0}}{V}\omega(p)g(p)}$(7)

Condition $g(0)>0$ that provides positivity of the value under the square root corresponds tothe domination of repulsion, cf. [6]7The Hamiltonian takes the form$\Gamma=-\frac{\lambda N_{0}^{2}}{2V}g(0)+\sum_{p\neq 0}E(p)a^{*}(p)a(p)$(8)

The state for hte new Hamiltonian (8) corresponds to astate of the non-ideal Bose gas
modified due to interaction.
Let us discuss the Bogoliubov-Landau condition for Hamiltonian the (8). We get for (1)$E(p)+ \frac{m(u-l)^{2}m}{2}-\frac{mu^{2}}{2}=E(p)-pu+\frac{p^{2}}{2m}$(9)Since $E(p)=c|p|$ for small we can get the situation when for $u<c$ the energy of excitation ispositive and transition from condensate to normal state proceeds with consumption of energy.
In this case we can apply the discussion of (2) which shows that the Bogoliubov-Landau
condition is applicable to (9).
$p$

5Condensate-normal phase interaction in the stochastic limit
In the present section we will discuss how to describe the Bogoliubov-Landau condition by thestochastic limit approach.
We will consider the stochastic limit of asystem with phase transition. We will see that thephase transition leads to arising of non-linear master and kinetic equations.This shows thatphase transition and the stochastic limit procedure do not commute: to investigate asystemwith phase transition by the stochastic limit we should describe first the phase transition.
We introduce the Hamiltonian for condensate-normal state interaction and investigate it in
the stochastic limit.
Consider asystem with two Bose fields (phases)$[c(p), c^{+}(p’)]--\delta(p-p’)$, $[a(k), a^{+}(k’)]=\delta(k-k’)$

We describe the condensate by the Bose field $c(p)$ (system degrees of freedom) labeled
velocity index with free Hamiltonian$p$$H_{c}= \int\epsilon(p)c^{*}(p)c(p)d^{3}p$with dispersion$\epsilon(p)=\frac{mp^{2}}{2}$and state$\langle c^{*}(p)c(p’)\rangle=N(p)\delta(p-p’)$This state describes the distribution of condensate over velocities

by a

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The second Bose field $a(p)$ (reservoir degrees of freedom) describes excitations of the nonideal Bose gas (normal phase) which diagonalize the interaction (considered in the previous
section). This excitations correspond to pairs of particles with opposite momenta and has free
Hamiltonian$H_{ns}= \int E(p)a^{*}(p)a(p)d^{3}p$The state of $a(p)$ is an equilibrium$\langle a^{*}(k)a(k’)\rangle=\delta(k-k’)\frac{1}{e^{\beta\omega(k)}-1}$

The total Hamiltonian will be(10)$H=H_{c}+H_{ns}+\lambda H_{I}$with the interaction.

$H_{I}= \int\int\overline{f(k,p)}c^{*}(p)c(p-k)a(k)d^{3}pd^{3}k+\mathrm{h}.\mathrm{c}$(11)where $f(u,p)$ is the form-factor (complex valued test function).
After this we apply to the Hamiltonian (10) the stochastic limit approach: by the stochastic
golden rule we construct the master equation for the density of particles in the condensate$c^{*}(p)c(p)$ and prove that the stochastic golden rule gives the Bogoliubov-Landau condition forthis system.
This approach is close to the discussion in Abrikosov-Gorkov-Dzialoshinskii book: Consider
the Bose liquid with zero temperature that flows with velocity . In the presence of friction in
the liquid there will be elementary excitations with different velocities. An analysis of these, [5], [9].
elementary excitations gives the Bogoliubov-Landau condition,
The friction is due to the interaction term (11). The rescaled free evolution of the interaction(11) (of the collective field) is equal to$\mathrm{v}$$\mathrm{c}\mathrm{f}[2]$$A_{\lambda}(p, k, t)= \frac{1}{\lambda}e^{-\not\simeq}.\cdot\lambda c^{+}(p)a(k)c(p-k)e^{-_{\lambda}^{t}\sim^{H}}tH\cdot.=\frac{1}{\lambda}c^{+}(p)a(k)c(p-k)e^{-itE(p,k)/\lambda^{2}}$$A_{\lambda}^{+}(p, k, t)= \frac{1}{\lambda}e^{tH}.\cdot\lambda\sim c^{+}(p-k)a^{+}(k)c(p)e^{-_{\lambda}^{t}\sim^{H}}\dot{.}=\frac{1}{\lambda}c^{+}(p-k)a^{+}(k)c(p)e^{itE(p,k)/\lambda^{2}}$where$E(p, k)=E(k)+\epsilon(p-k)-\epsilon(p)$

(12)
(13)

(14)

is the corresponding energy difference.
The difference of energies (14) coincides with the difference of energies in the Bogoliubov
Landau condition (9) that justify the fact that Hamiltonian (10) describe the non-ideal Bosegas considered above.
The stochastic limit of the rescaled evolution of the interaction (12) gives rise to aquantum
white noise (master field):$\lim_{\lambdaarrow 0}A_{\lambda}(p, k, t)=B(p, k, t)$9After the stochastic limit we cannot separate in the master field the degrees of freedom ofreservoir and normal state. This means that the degrees of freedom of the condensate and of
the normal phase become entangled even at kinematical level, cf. [10].
The master field $B(p, k, t)$ will satisfy the variant of quantum Boltzmann statistics given by
the following theorem

Tie entangled noise algebra is generated by $B_{1}(p, k, t)$ , $B_{2}(p, k, t)$ , $n(p)$ where

Theorem 1.

$B(p, k, t)=B_{1}(p, k, t)+B_{2}^{*}(p, k, t)$with relations$B_{1}B_{2}^{*}=B_{2}B_{1}^{*}=0$$B_{1}(p, k, t)B_{1}^{*}(p’, k’, t’)=2 \pi\delta(t-t’)\delta(E(p, k))n(p)(N(p-k)+1)\delta(p-p’)\frac{\delta(k-k’)}{1-e^{-\beta\omega(k)}}$(15)$B_{2}(p, k, t)B_{2}^{*}(p’, k’, t’)=2 \pi\delta(t-t’)\delta(E(p, k))n(p-k)(N(p)+1)\delta(p-p’)\frac{\delta(k-k’)}{e^{\beta\omega(k)}-1}$(16)$[n(p’), B_{1}^{\mp}(p, k, t)]=(\pm)(\delta(p’-p)-\delta(p’-p+k))B_{1}^{\mp}(p, k, t)$$[n(p’), B_{2}^{\mp}(p, k, t)]=(\mp)(\delta(p’-p)-\delta(p’-p+k))B_{2}^{\mp}(p, k, t)$$[n(p), n(p’)]=0$

Here$B_{i}^{-}=B_{i}$,$B_{i}^{+}=B_{i}^{*}$.

The zero temperature version of this theorem was obtained in [10]. The one particle sector
relations $(n(p)=1)$ were investigated in [11], [12], [13].
Consider now the evolution in the stochastic limit for the considered Hamiltonian. By thestochastic golden rule, cf. [7] we get
Theorem 2.

The stochastic differential equation for$U_{t}$

has the form

$dU_{t}=(-i \int dkdp$ $(\overline{f}(k,p)dB(p, k, t)+dB^{\dagger}(p, k, t)f(k,p))-$$-dt(f|f)_{-}-dt\overline{(f|f)}_{+})U_{t}$(17)where the stochastic diferentials satisfy the $Ito$ table$dB(p, k, t)dB^{\uparrow}(p’, k’, t)=2\pi n(p)\delta(\omega(k)+\epsilon(p-k)-\epsilon(p))$$(N(p-k)+1) \delta(p-p’)\frac{\delta(k-k’)}{1-e^{-\beta\omega(k)}}dt$(18)$dB^{\dagger}(p, k, t)dB(p’, k’, t)=2\pi n(p-k)\delta(\omega(k)+\epsilon(p-k)-\epsilon(p))$$(N(p)+1) \delta(p-p’)\frac{\delta(k-k’)}{e^{\beta\omega(k)}-1}dt$

(19)

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$(f|f)_{-}= \int dkdp|f(k,p)|^{2}\frac{-in(p)(N(p-k)+1)}{\omega(k)+\epsilon(p-k)-\epsilon(p)-i0}\frac{1}{1-e^{-\beta\omega(k)}}$$(f|f)_{+}= \int dkdp|f(k,p)|^{2}\frac{-in(p-k)(N(p)+1)}{\omega(k)+\epsilon(p-k)-\epsilon(p)-i0}\frac{1}{e^{\beta\omega(k)}-1}$

One particle version of this quantum stochastic differential equation was obtained and investigated in [14].

6Master equation and superfluidity
In the present section we consider the master equation for the stochastic dynamics described in
the previous section. The master equation is the equation for the expectation over the degrees
of freedom of the reservoir (normal state in the considered case) of the Heisenberg dynamics
for some observable $X$$\frac{d}{dt}X_{t}=\frac{d\langle U_{t}^{*}XU_{t}\rangle}{dt}=\frac{d}{dt}\langle j_{t}(X)\rangle=\langle j_{t}(\theta_{0}(X))\rangle$(20)

defined by theorem 2.
Consider the master equation for the density $n(p)=c^{*}(p)c(p)$ . The stochastic golden rule(application of theorems 2and 1) giveswith$U_{t}$$\frac{d}{dt}n_{t}(q)=\int dkdp|f(k,p)|^{2}(\delta(q-p)-\delta(q-p+k))2\pi\delta(E(k)+\epsilon(p-k)-\epsilon(p))$$(n_{t}(p-k)(N(p)+1) \frac{1}{e^{\beta E(k)}-1}-n_{t}(p)(N(p-k)+1)\frac{1}{1-e^{-\beta E(k)}})$(21)

is in the abelian subalgebra this equation is classical and we can consider $n(p)$ as a
Sinceclassical distribution.
Taking the integral over one gets$n_{t}(p)$$p$$\frac{d}{dt}n_{t}(q)=2\pi\int dk(|f(k, q|^{2}\delta(E(k)+\epsilon(q-k)-\epsilon(q))$$(n_{t}(q-k)(N(q)+1) \frac{1}{e^{\beta E(k)}-1}-n_{t}(q)(N(q-k)+1)\frac{1}{1-e^{-\beta E(k)}})-$$-|f(k, q+k)|^{2}\delta(E(k)+\epsilon(q)-\epsilon(q+k))$$(n_{t}(q)(N(q+k)+1) \frac{1}{e^{\beta E(k)}-1}-n_{t}(q+k)(N(q)+1)\frac{1}{1-e^{-\beta E(k)}}))$(22)

Let us discuss the connection of the Bogoliubov-Landau condition with (22). Since the disperwe get
is radiative $E(k)=c|k|$ andsion (7) for small$|k|$$\epsilon(p)=L^{2}2m$,$E(k)+ \epsilon(q-k)-\epsilon(q)=c|k|-\frac{1}{m}qk+\frac{k^{2}}{2m}>0$$\frac{1}{m}|q|<c$11$E(k)+ \epsilon(q)-\epsilon(q+k)=c|k|-\frac{1}{m}qk-\frac{k^{2}}{2m}$. Let us note that the second
This implies that the first -function in (22) vanishes for-function in (22) does not vanish for any .
To get asuperfluid motion in (22) it is sufficient to consider such $n(q)$ that the following
products will vanish for arbitrary and :$\frac{1}{m}|q|<c$$\delta$$q$$\delta$$q$$k$$n(q)\delta(E(k)+\epsilon(q-k)-\epsilon(q))=0$(23)$n(q-k)\delta(E(k)+\epsilon(q-k)-\epsilon(q))=0$(24)

If $n(q)$ would satisfy (23), (24) then the RHS of (22) would vanish.
To satisfy (23) it is sufficient to take the support of $n(q)$ is concentrated in the ball $|q|\leq mc$ .
But (24) can not be satisfied when the support of $n(q)$ is concentrated in the ball $|q|\leq mc$ .
This implies that (22) does not describe asuperfluid liquid and to investigate superfluidity inthe stochastic limit one needs some additional arguments.
To overcome this problem we propose the following construction.
First, since $N(p)>>1$ we can identify $N$ and $N+1$ .
with expectation $\langle n(p)\rangle=N(p)$ in (22) (weSecond (and most important), we identify
in this formula). The identification
substitute $N(p)$ by$n_{t}(p)$$n_{t}(p)$$\langle n(p)\rangle=N(p):=n_{t}(p)$(25)

means that we consider the stochastic limit with the state depending on time in the slow timescale of the stochastic limit. This modifies the stochastic limit procedure in order to take into
account the effects of phase transition.
This condition may be justified by the following argument. Equation (20) for $X=n(q)$ is$\frac{d}{dt}\langle U_{t}^{*}n(q)U_{t}\rangle=\langle j_{t}(\theta_{0}(n(q)))\rangle$(26)

is equal to the RHS of (22). Then we apply the identification (25) for $t=0$ to
where(26). We get, in the integral on RHS of (26), combinations of terms of the Mowing form$\theta_{0}(n(q))$$j_{t}(n(q)n(q-k))=j_{t}(n(q))j_{t}(n(q-k))$since$j_{t}$

is ahomomorphism. After this we apply the semiclassical approximation$\langle j_{t}(n(q))j_{t}(n(q-k))\rangle=\langle j_{t}(n(q))\rangle\langle j_{t}(n(q-k))\rangle=n_{t}(q)n_{t}(q-k)$that justifies (25) for any .
Condition (25) introduces the self-interaction into the considered model and makes equation(22) nonlinear. Condition (25) implies for (22) the following:$t$$\frac{d}{dt}n_{t}(q)=-2\pi\int dk(|f(k, q|^{2}\delta(E(k)+\epsilon(q-k)-\epsilon(q))n_{t}(q)n_{t}(q-k)-$$-|f(k, q+k)|^{2}\delta(E(k)+\epsilon(q)-\epsilon(q+k))n_{t}(q)n_{t}(q+k))$

(27)

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Let us note that this equation is nonlinear (quadratic).
For (27) the condition of superfluidity is reduced to (23) (condition (24) can be ignored).
This implies that (27) describes asuperfluid flow. We get that in the stochastic limit thecondition
Supp $n(q)\subset\{q : |q|\leq mc\}$(28)

is sufficient for superfluidity in the sence that, under this condition, the right hand side of (27)is zero and the density, of the condensate, is constant.
We call equation (27) the equation of motion of superfluid liquid.
Non-linear master equation (27) is an example of ageneral phenomenon. Application of thestochastic limit to systems with phase transitions will generally create non-linear master andkinetic equations. Non-linearity will enter through the self-interaction given by an analogueof (25) for acertain phase (an analogue of the condensate considered in the present paper).
$n_{t}(q)$7Conclusion
We summarize our main results as follows:
1) The Bogoliubov-Landau condition of superfluidity follows from the stochastic limit approach.
2) The stochastic limit approach gives not only acondition of superfluidity but also anatural
candidate for the equation of motion of the superfluid liquid (27).
3) This equation is nonlinear (quadratic).
4) Without the introduction of anon-linearity (identification of $n(p)$ and $N(p)$ ) it is impossible
to get superfluidity in the stochastic limit approach.

5) Nonlinear master equations should be ageneral feature of the stochastic limit of selfinteracting systems.

Acknowledgements
The authors are grateful to I.V.Volovich and H.-D.Doebner for discussions. Sergei Kozyrev
is grateful to Centro Vito Volterra and Luigi Accardi for kind hospitality. This work waspartially supported by INTAS 9900545 grant. Sergei Kozyrev was partially supported by RFFI
990100866 grant.

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