# Two-fluid hydrodynamics of cold atomic bosons under influence of the quantum
fluctuations at non-zero temperatures

Pavel A. Andreev<EMAIL_ADDRESS>Faculty of physics, Lomonosov
Moscow State University, Moscow, Russian Federation, 119991.  Peoples
Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street,
Moscow, 117198, Russian Federation

###### Abstract

Ultracold Bose atoms is the physical system, where the quantum and nonlinear
phenomena play crucial role. Ultracold bosons are considered at the small
finite temperatures. Bosons are considered as two different fluids: Bose-
Einstein condensate and normal fluid (the thermal component). An extended
hydrodynamic model is obtained for both fluids, where the pressure evolution
equations and the pressure flux third rank tensor evolution equations are
considered along with the continuity and Euler equations. It is found that the
pressure evolution equation contains zero contribution of the short-range
interaction. The pressure flux evolution equation contains the interaction
which gives the quantum fluctuations in the zero temperature limit. Here, we
obtain its generalization for the finite temperature. The contribution of
interaction in the pressure flux evolution equation which goes to zero in the
zero temperature limit is found. The model is obtained via the straightforward
derivation from the microscopic many-particle Schrodinger equation in the
coordinate representation.

BEC, pressure evolution equation, hydrodynamics, finite temperatures, quantum
fluctuations.

###### pacs:

03.75.Hh, 03.75.Kk, 67.85.Pq

##  I Introduction

Small temperature bosons are studied in terms of two-fluid hydrodynamics
consisting of the Bose-Einstein condensate (BEC) and normal fluid Dalfovo RMP
99 . Each fluid is considered in terms of two hydrodynamic equations: the
continuity and Euler equations. It is assumed that the BEC can be completely
described by the concentration and velocity field, or, in other terms, by the
Gross-Pitaevskii equation Dalfovo RMP 99 , since BEC is the collection of
particles in the single quantum state. However, the normal fluid model
requires an truncation of the set of hydrodynamic equations. The pressure of
normal fluid existing in the Euler equation for the normal fluid is an
independent function. Equation for the pressure evolution provides an
expression for the pressure perturbations via the perturbations of other
functions. Application of the equation of state for pressure makes the model
more simple, but equation of state for pressure leads to the less accurate
model.

Moreover, the kinetic pressure in the Euler equation for the BEC is usually
chosen to be equal to zero Andreev PRA08 , Andreev LP 19 . Since the kinetic
pressure is related to the occupation of the excited states However, there is
the nonzero part caused by the quantum fluctuations Andreev 2005 . The
pressure evolution equation of the weakly interacting bosons contains no
interaction, but next equation in the chain of the quantum hydrodynamic
equations (the equation for the pressure flux third rank tensor) contains the
interaction causing the depletion of the BECs at the zero temperature.
Therefore, it is necessary to consider the pressure flux evolution equation
both for the BEC and for the normal fluid at the analysis of the small
temperature influence.

The quantum depletion of the BECs is the appearance of the bosons in the
excited states while system is kept at the zero temperature. So, some energy
of the collective motion is transferred to the individual motion of a portion
of particles. It is caused by the quantum fluctuations related to the
interparticle interaction. The quantum fluctuations are considered in
literature for a long time. Mostly, their theoretical analysis is based on the
Bogoliubov-de Gennes approach Lee PR 57 , Pitaevskii PRL 98 , Braaten PRL 99 ,
Astrakharchik PRL 05 . The quantum fluctuations in BECs are studied
experimentally as well Xu PRL 06 , Altmeyer PRL 07 , Papp PRL 08 . This method
is generalized for the dipolar BECs, where the quantum fluctuations plays
crucial role at the description of the dipolar BECs of lantanoids. The dipolar
lantanoid BECs reveal the large scale instability causing the splitting of the
cloud of atoms on the number of macroscopic drops. This highly nonlinear
phenomena is called the quantum droplet formation Kadau Pfau Nature 16 ;
Ferrier-Barbut PRL 16 ; Baillie PRA 16 ; Bisset PRA 16 ; Wachtler PRA 16 a1 ;
Wachtler PRA 16 a2 ; Blakie pra 16 ; Boudjemaa PRA 20 ; Heinonen PRA 19 ;
Malomed Phys D 19 ; Shamriz PRA 20 ; Li PRA 19 ; Aybar PRA 19 ; Examilioti JP
B 20 ; Miyakawa PRA 20 ; Bottcher arXiv 20 07 ; Bisset arXiv 20 07 ; Wang
arXiv 20 02 ; Edmonds arXiv 20 02 ; Baillie PRA 20 . Therefore, reassemble of
bosons in smaller compact groups causes the stabilization of the system. These
studies give the motivation for the study of quantum fluctuations. However,
here we restrict our analysis with the short-range interaction only and no
dipole-dipole interaction is discussed. Moreover, the influence of the finite
temperature is the necessary part of complete model of these phenomena.

Fundamental feature of the collective dynamics in the spectrum of the sound
waves. Two distinct sound velocities exist in finite temperature ultracold
Bose gas Dalfovo RMP 99 , Griffin PRB 96 . The two-fluid model shows that the
slower mode (second sound) is associated with the BEC component, while the
faster mode (first sound) is associated with the thermal component.
Generalized expressions for the speeds of sounds are obtained within developed
model.

Derivation of two-fluid hydrodynamics for the finite temperature bosons in the
limit of small temperature, where the large fraction of the bosons is located
in the BEC state is given from the microscopic motion in accordance with the
quantum hydrodynamic method Andreev PRA08 , Andreev LP 19 , MaksimovTMP 2001 ,
Andreev PTEP 19 . The microscopic dynamics is described by the Schrodinger
equation in the coordinate representation. Collection of the macroscopic
functions is presented to describe the collective effects in ultracold bosons.
The last includes the concentration of particles, the velocity field and the
pressure tensor. The derivation of basic equations is made for all bosons
distributed on the lower energy level and the excited levels as the single
fluid. The decomposition on two fluids is made on the microscopic scale. After
general structure of equations is obtained for the arbitrary temperature and
arbitrary strength of interaction, an approximate calculation of functions
presenting the interaction is made for the regime of short-range interaction.
Hence, the small parameter related to the small area of interaction potential
is used. It gives a specification for general model, but also the first order
contribution on the small parameter is applied. The further truncation is made
at calculation of the interaction terms for weak interaction and small
temperatures.

This paper is organized as follows. In Sec. II major steps of derivation of
hydrodynamic equations from the Schrodinger equation are demonstrated, where
the pressure evolution equation (the quantum Bohm potential evolution
equation) and the third rank tensor evolution equation are obtained along with
the continuity and Euler equations. In Sec. III calculation of interaction in
the Euler equation, the pressure evolution equation, and the pressure flux
evolution equation is demonstrated. In Sec. IV presents the suggested version
of the extended two-fluid quantum hydrodynamic model for the ultracold finite
temperature bosons. In Sec. V the limiting regime of derived model for the BEC
is obtained under influence of the quantum fluctuations at the zero
temperature. In Sec. VI a brief summary of obtained results is presented.

##  II On derivation of hydrodynamic equations from microscopic quantum
dynamics

###  II.1 Basic definitions of quantum hydrodynamics and the Euler equation
derivation

On the microscopic level we do not have notion of temperature. Hence we
consider system of interacting bosons governed by the Schrodinger equation
$\imath\hbar\partial_{t}\Psi=\hat{H}\Psi$ with the following Hamiltonian

$\hat{H}=\sum_{i=1}^{N}\biggl{(}\frac{\hat{\textbf{p}}^{2}_{i}}{2m_{i}}+V_{ext}(\textbf{r}_{i},t)\biggr{)}+\frac{1}{2}\sum_{i,j\neq
i}U(\textbf{r}_{i}-\textbf{r}_{j}),$ (1)

where $m_{i}$ is the mass of i-th particle,
$\hat{\textbf{p}}_{i}=-\imath\hbar\nabla_{i}$ is the momentum of i-th
particle. The last term in the Hamiltonian (1) is the boson-boson interaction
$U_{ij}$. We do not specify the form of interaction. However, the derivation
presented below employs that the interaction has finite value on the small
distances between particles and shows fast decay at the increase of the
interparticle distance. Definitely, no distinguishing between bosons in the
BEC state and bosons in other states is made at this stage. Separation of all
bosons on two subsystems is made in terms of collective variables.

Hydrodynamic model usually made for each species of particles. If we consider
a single species then all masses equal to each other.

Distribution of particles in a trap, waves, solitons, oscillations of form of
trapped particles are described by the concentration of particles.
Concentration is an essential macroscopic function both for classical and for
quantum fluids. The module of the macroscopic wave function in the Gross-
Pitaevskii equation gives the square root of concentration of bosons in the
BEC state. Therefore, we start the derivation of quantum hydrodynamic
equations from the definition of concentration. The quantum mechanics is based
on notion of point-like objects in spite the wave nature of quantum objects.
So, the eigenfunction of the coordinate operator in the coordinate
representation is the delta function
$\hat{x}\psi_{x^{\prime}}(x)=x^{\prime}\psi_{x^{\prime}}(x)$, where normalized
wave function is $\psi_{x^{\prime}}(x)=\delta(x-x^{\prime})$. Obviously, the
operation of concentration in the coordinate representation of quantum
mechanics is the sum of delta functions
$\hat{n}=\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})$. Moreover, it is
supported by the general rule for quantization. We need to take corresponding
classical function.

Transition to description of the collective motion of bosons is made via
introduction of the concentration Andreev PRA08 , MaksimovTMP 2001 :

$n=\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\Psi^{*}(R,t)\Psi(R,t),$ (2)

which is the first collective variable in our model. Other collective
variables appear during the derivation. Equation (2) contains the following
notations $dR=\prod_{i=1}^{N}d\textbf{r}_{i}$ is the element of volume in $3N$
dimensional configurational space, with $N$ is the number of bosons.
Concentration (2) is the sum of partial concentrations $n=n_{n}+n_{b}$
describing the distribution of BEC $n_{b}$ and normal fluid $n_{n}$ in the
coordinate space.

The equation for evolution of concentration (2) can be obtained by acting by
time derivative on function (2). The time derivative acts on the wave
functions under the integral while the time derivatives of the wave function
are taken from the Schrodinger equation. Obtain the continuity equation for
concentration (2) after straightforward calculations

$\partial_{t}n+\nabla\cdot\textbf{j}=0,$ (3)

where the new collective function called the current appears as the following
integral of the wave function

$\textbf{j}(\textbf{r},t)=\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\times$
$\times\frac{1}{2m_{i}}(\Psi^{*}(R,t)\hat{\textbf{p}}_{i}\Psi(R,t)+c.c.),$ (4)

with $c.c.$ is the complex conjugation.

Both introduced collective functions $n(\textbf{r},t)$ and
$\textbf{j}(\textbf{r},t)$ are quadratic forms of the wave function. Each of
them can be splitted on two parts related to the BEC and normal fluid. Hence
we have $n=n_{n}+n_{b}$ and $\textbf{j}=\textbf{j}_{n}+\textbf{j}_{b}$. No
microscopic definitions are introduced for the partial functions $n_{n}$,
$n_{b}$, $\textbf{j}_{n}$, and $\textbf{j}_{b}$. Therefore, the continuity
equation (3) splits on two partial continuity equations

$\partial_{t}n_{a}+\nabla\cdot\textbf{j}_{a}=0,$ (5)

where subindex $a$ stands for $b$ and $n$.

Continue the derivation of hydrodynamic equations and consider the time
evolution of the particle current (4). Act by time derivative on function j
(4) and use the Schrodinger equation with Hamiltonian (1). It leads to the
general form of the current evolution equation

$\partial_{t}j^{\alpha}+\partial_{\beta}\Pi^{\alpha\beta}=-\frac{1}{m}n\partial_{\alpha}V_{ext}+\frac{1}{m}F^{\alpha}_{int},$
(6)

where

$\Pi^{\alpha\beta}=\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{4m^{2}}[\Psi^{*}(R,t)\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi(R,t)$
$+\hat{p}_{i}^{\alpha*}\Psi^{*}(R,t)\hat{p}_{i}^{\beta}\Psi(R,t)+c.c.]$ (7)

is the momentum flux, and

$F^{\alpha}_{int}=-\int(\partial^{\alpha}U(\textbf{r}-\textbf{r}^{\prime}))n_{2}(\textbf{r},\textbf{r}^{\prime},t)d\textbf{r}^{\prime},$
(8)

with the two-particle concentration

$n_{2}(\textbf{r},\textbf{r}^{\prime},t)$ $=\int dR\sum_{i,j=1,j\neq
i}^{N}\delta(\textbf{r}-\textbf{r}_{i})\delta(\textbf{r}^{\prime}-\textbf{r}_{j})\Psi^{*}(R,t)\Psi(R,t).$
(9)

It is necessary to split equation (6) on two equations for each subsystem of
bosons. In current form equation (6) consist of superposition of functions
which are quadratic forms of the wave function. Hence, each term can be
splitted on two parts and we find two similar equations for the currents

$\partial_{t}j_{a}^{\alpha}+\partial_{\beta}\Pi_{a}^{\alpha\beta}=-\frac{1}{m}n_{a}\partial_{\alpha}V_{ext}+\frac{1}{m}F^{\alpha}_{a,int}.$
(10)

The first and third terms are proportional to the concentration and the
current. therefore, they require no comments. Nontrivial difference between
two current evolution equation appears at further analysis of the momentum
flux $\Pi^{\alpha\beta}$ and the interaction $F^{\alpha}_{int}$. However, we
point out some difference which appear for the momentum flux
$\Pi^{\alpha\beta}$. Its structure is obtained in many papers (see for
instance Andreev PRA08  after equation (52), Andreev 2001  equation (24))

$\Pi^{\alpha\beta}=nv^{\alpha}v^{\beta}+p^{\alpha\beta}+T^{\alpha\beta},$ (11)

where $p^{\alpha\beta}$ is the pressure tensor, and $T^{\alpha\beta}$ is the
tensor giving the quantum Bohm potential, its approximate form can be written
in the following form

$T^{\alpha\beta}_{0}=-\frac{\hbar^{2}}{4m^{2}}\biggl{[}\partial_{\alpha}\partial_{\beta}n-\frac{\partial_{\alpha}n\cdot\partial_{\beta}n}{n}\biggr{]}.$
(12)

Tensor $T^{\alpha\beta}_{0}$ (12) is obtained for noninteracting particles
located in the single quantum state.

Basically, the pressure tensor $p^{\alpha\beta}$ is defined via the wave
function $\Psi(R,t)$. However, it requires some manipulations with the wave
function and introduction of a number of intermediate function. Hence we do
not present its explicit form. Nevertheless, the pressure tensor is related to
the distribution of bosons on quantum states with energies above $E_{min}$.
Therefore, for bosons in the BEC state we have $p^{\alpha\beta}_{B}=0$ if no
quantum fluctuations are considered and

$\Pi_{B}^{\alpha\beta}=n_{B}v_{B}^{\alpha}v_{B}^{\beta}+T_{B}^{\alpha\beta}+p_{qf}^{\alpha\beta},$
(13)

where $T_{B}^{\alpha\beta}$ is the function of $n_{B}$ (12) if there is no
interaction. Distribution of particles on different quantum states does not
allow to get full expression (12), but the first term. However, it can be used
as an equation of state for noninteracting limit. The normal fluid bosons have
nonzero pressure $p_{n}^{\alpha\beta}\neq 0$. Hence, all terms in presentation
(11) exists in this regime. Expression (12) appears for bosons in the single
state in the absence of interaction. Hence, it is an approximate expression
for the weakly interacting bosons being in the BEC state. It is even less
accurate for normal fluid bosons, but we use it as an equation of state for
the quantum part of the momentum flux.

###  II.2 The pressure evolution equation

Extending the set of hydrodynamic equations we can derive the equation for the
momentum flux evolution. It can be expected that this equation brings extra
information for the normal fluid bosons only. However, the quantum
fluctuations give contribution in the evolution of the kinetic pressure of
BECs in the limit of zero temperature via the divergence of the third rank
tensor. If the temperature is nonzero we have two partial kinetic pressures
for the BEC and for the normal fluid. Consider the time evolution of the
momentum flux (7) using the Schrodinger equation with Hamiltonian (1).

It gives to the following expression:

$\partial_{t}\Pi^{\alpha\beta}=\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{4m^{2}}[\hat{H}^{*}\Psi^{*}(R,t)\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi(R,t)$
$-\Psi^{*}(R,t)\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{H}\Psi(R,t)+\hat{p}_{i}^{\alpha*}\hat{H}^{*}\Psi^{*}(R,t)\hat{p}_{i}^{\beta}\Psi(R,t)$
$-\hat{p}_{i}^{\alpha*}\Psi^{*}(R,t)\hat{p}_{i}^{\beta}\hat{H}\Psi(R,t)-c.c.]$
(14)

The part of the presented terms contains the Hamiltonian $\hat{H}$ under
action of the momentum operators. We permute the Hamiltonian $\hat{H}$ and the
operators acting on it. Hence, the result of permutation presented by terms
where no operators act on the Hamiltonian $\hat{H}$. However, the terms
containing the corresponding commutators appear. Therefore, all terms are
combined in two groups:

$\partial_{t}\Pi^{\alpha\beta}=\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{4m^{2}}[\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi$
$-\Psi^{*}\hat{H}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi+\hat{H}^{*}\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\Psi-\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot\hat{H}\hat{p}_{i}^{\beta}\Psi-c.c.]$
$+\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{4m^{2}}[-\Psi^{*}[\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta},\hat{H}]\Psi$
$+[\hat{p}_{i}^{\alpha*},\hat{H}^{*}]\Psi^{*}\cdot\hat{p}_{i}^{\beta}\Psi-\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot[\hat{p}_{i}^{\beta},\hat{H}]\Psi-c.c.]$
(15)

The first group of terms in expression (15) gives the divergence of flux of
tensor $\Pi^{\alpha\beta}$. The second group of terms contains the
commutators. This group leads to the contribution of interaction in the
momentum flux evolution.

It gives the momentum flux evolution equation

$\partial_{t}\Pi^{\alpha\beta}+\partial_{\gamma}M^{\alpha\beta\gamma}=-\frac{1}{m}j^{\beta}\partial_{\alpha}V_{ext}$
$-\frac{1}{m}j^{\alpha}\partial_{\beta}V_{ext}+\frac{1}{m}(F^{\alpha\beta}+F^{\beta\alpha}),$
(16)

where the momentum flux is the full flux of all bosons
$\Pi^{\alpha\beta}=\Pi_{n}^{\alpha\beta}+\Pi_{b}^{\alpha\beta}$, the splitting
on two subspecies is to be made later,

$F^{\alpha\beta}=-\int[\partial^{\alpha}U(\textbf{r}-\textbf{r}^{\prime})]j_{2}^{\beta}(\textbf{r},\textbf{r}^{\prime},t)d\textbf{r}^{\prime}$
(17)

is the force tensor field,

$M^{\alpha\beta\gamma}=\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{8m_{i}^{3}}\biggl{[}\Psi^{*}(R,t)\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi(R,t)$
$+\hat{p}_{i}^{\alpha*}\Psi^{*}(R,t)\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi(R,t)+\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}(R,t)\hat{p}_{i}^{\beta}\Psi(R,t)$
$+\hat{p}_{i}^{\gamma*}\Psi^{*}(R,t)\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi(R,t)+c.c.\biggr{]}$
(18)

is the current (flux) of the momentum flux, and

$\textbf{j}_{2}(\textbf{r},\textbf{r}^{\prime},t)=\int dR\sum_{i,j\neq
i}\delta(\textbf{r}-\textbf{r}_{i})\delta(\textbf{r}^{\prime}-\textbf{r}_{j})\times$
$\times\frac{1}{2m_{i}}[\Psi^{*}(R,t)\hat{\textbf{p}}_{i}\Psi(R,t)+c.c.]$ (19)

is the two-particle current-concentration function.

If quantum correlations are dropped function
$j_{2}^{\alpha}(\textbf{r},\textbf{r}^{\prime},t)$ splits on product of the
current $j^{\alpha}(\textbf{r},t)$ and the concentration
$n(\textbf{r}^{\prime},t)$. Interaction in the momentum flux evolution
equation (16) is presented by symmetrized combinations of tensors
$F^{\alpha\beta}$, which is the flux or current of force.

Partial momentum flux equations appear as

$\partial_{t}\Pi_{a}^{\alpha\beta}+\partial_{\gamma}M_{a}^{\alpha\beta\gamma}=-\frac{1}{m}j_{a}^{\beta}\partial_{\alpha}V_{ext}$
$-\frac{1}{m}j_{a}^{\alpha}\partial_{\beta}V_{ext}+\frac{1}{m}(F_{a}^{\alpha\beta}+F_{a}^{\beta\alpha}),$
(20)

where
$M^{\alpha\beta\gamma}=M_{B}^{\alpha\beta\gamma}+M_{n}^{\alpha\beta\gamma}$,
with

$M_{a}^{\alpha\beta\gamma}=n_{a}v_{a}^{\alpha}v_{a}^{\beta}v_{a}^{\gamma}+v_{a}^{\alpha}(p_{a}^{\beta\gamma}+T_{a}^{\beta\gamma})+v_{a}^{\beta}(p_{a}^{\alpha\gamma}+T_{a}^{\alpha\gamma})$
$+v_{a}^{\gamma}(p_{a}^{\alpha\beta}+T_{a}^{\alpha\beta})+Q_{a}^{\alpha\beta\gamma}+T_{a}^{\alpha\beta\gamma}+L_{a}^{\alpha\beta\gamma}.$
(21)

The pressure is the average of the square of the thermal velocity, when tensor
$Q_{a}^{\alpha\beta\gamma}$ is the average of the product of three projections
of the thermal velocity. Function $L_{a}^{\alpha\beta\gamma}$ presents
quantum-thermal terms. For the BEC we have $p_{B}^{\alpha\beta}=0$,
$Q_{B}^{\alpha\beta\gamma}=0$, $L_{B}^{\alpha\beta\gamma}=0$, since it has no
contribution of the excited states. For symmetric equilibrium distributions we
have $Q_{n}^{\alpha\beta\gamma}=0$, $L_{n}^{\alpha\beta\gamma}=0$. We
generalize this conclusion for nonequilibrium states as the trivial equations
of state for these functions. Tensor $T_{a}^{\alpha\beta\gamma}$ is

$T_{a}^{\alpha\beta\gamma}=-\frac{\hbar^{2}}{12m^{2}}n_{a}(\partial^{\alpha}\partial^{\beta}v_{a}^{\gamma}+\partial^{\alpha}\partial^{\gamma}v_{a}^{\beta}+\partial^{\beta}\partial^{\gamma}v_{a}^{\alpha}).$
(22)

This definition of tensor $T^{\alpha\beta\gamma}$ differs from equation (27)
in Ref. Andreev 2001  by extraction of the quantum Bohm potentials written
together with pressure tensors in equation (21). Equation (27) in Ref. Andreev
2001  contains approximate form of the quantum Bohm potential
$T^{\alpha\beta}$. Equation (21) includes the quantum Bohm potential in its
general form. Moreover, expression (22) is an exact formula obtained with no
assumption about structure of the many-particle wave function like the first
term in equation (23) in Ref. Andreev 2001 .

Equations (3)-(20) are obtained in general form. The short-range nature of the
inter-particle interaction is not used. Moreover, the traditional hydrodynamic
equations are presented via the velocity field and the pressure tensor while
equations (3)-(20) are written via the current and the momentum flux.

The method of the introduction of the velocity field in the equations of
quantum hydrodynamics of spinless particles is presented in Refs. Andreev
PRA08 , Andreev 2001 . The method of calculation of the terms containing
interaction for the short-range interaction limit is also described in Refs.
Andreev PRA08 , Andreev 2001 . Let us present results of application of these
methods for finite temperature bosons. Moreover, we consider the short-range
interaction in the first order by the interaction radius.

###  II.3 Appearance of the quantum fluctuations in the third rank tensor
evolution equation

Derivation of the quantum fluctuations requires the calculation of the time
evolution of the current of the momentum flux $M^{\alpha\beta\gamma}$ (18).
The method of derivation is similar to the equations obtained above. The time
derivative of tensor $M^{\alpha\beta\gamma}$ acts on the wave function in its
definition. The time derivative of the wave function is replaced by the
Hamiltonian (1) in accordance with the many-particle microscopic Schrodinger
equation $\imath\hbar\partial_{t}\Psi=\hat{H}\Psi$. It leads to the following
expression:

$\partial_{t}M^{\alpha\beta\gamma}=\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{8m_{i}^{3}}\biggl{[}\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi$
$-\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\hat{H}\Psi+\hat{p}_{i}^{\alpha*}\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi+\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\Psi$
$-\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\hat{H}\Psi-\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\hat{H}\Psi$
$+\hat{p}_{i}^{\gamma*}\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi-\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{H}\Psi-c.c.\biggr{]}$
(23)

The part of the presented terms contains the Hamiltonian $\hat{H}$ under
action of the momentum operators. We permute the Hamiltonian $\hat{H}$ and the
operators acting on it. Hence, the result of permutation presented by terms
where no operators act on the Hamiltonian $\hat{H}$. However, the terms
containing the corresponding commutators appear. Therefore, all terms are
combined in two groups:

$\partial_{t}M^{\alpha\beta\gamma}=\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{8m_{i}^{3}}\biggl{[}\hat{H}^{*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi$
$-\Psi^{*}\hat{H}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi+\hat{H}^{*}\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi+\hat{H}^{*}\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{p}_{i}^{\beta}\Psi$
$-\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot\hat{H}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi-\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{H}\hat{p}_{i}^{\beta}\Psi$
$+\hat{H}^{*}\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi-\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot\hat{H}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi-c.c.\biggr{]}$

$+\frac{\imath}{\hbar}\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{8m_{i}^{3}}\biggl{[}-\Psi^{*}[\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma},\hat{H}]\Psi$
$+[\hat{p}_{i}^{\alpha*},\hat{H}^{*}]\Psi^{*}\cdot\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi+[\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*},\hat{H}^{*}]\Psi^{*}\cdot\hat{p}_{i}^{\beta}\Psi$
$-\hat{p}_{i}^{\alpha*}\Psi^{*}\cdot[\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma},\hat{H}]\Psi-\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot[\hat{p}_{i}^{\beta},\hat{H}]\Psi$
$+[\hat{p}_{i}^{\gamma*},\hat{H}^{*}]\Psi^{*}\cdot\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi-\hat{p}_{i}^{\gamma*}\Psi^{*}\cdot[\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta},\hat{H}]\Psi-c.c.\biggr{]}.$
(24)

The first group of terms leads to the divergence of the flux of tensor
$M^{\alpha\beta\gamma}$. The second group of terms containing the commutators
presents the interactions.

Final form of tensor $M^{\alpha\beta\gamma}$ evolution equation can be
expressed in the following terms:

$\partial_{t}M^{\alpha\beta\gamma}+\partial_{\delta}R^{\alpha\beta\gamma\delta}=\frac{\hbar^{2}}{4m^{3}}n\partial_{\alpha}\partial_{\beta}\partial_{\gamma}V_{ext}$
$-\frac{1}{m}\Pi^{\beta\gamma}\partial_{\alpha}V_{ext}-\frac{1}{m}\Pi^{\alpha\gamma}\partial_{\beta}V_{ext}-\frac{1}{m}\Pi^{\alpha\beta}\partial_{\gamma}V_{ext}$
$+\frac{1}{m}F_{qf}^{\alpha\beta\gamma}+\frac{1}{m}(F^{\alpha\beta\gamma}+F^{\beta\gamma\alpha}+F^{\gamma\alpha\beta}),$
(25)

where

$F_{qf}^{\alpha\beta\gamma}=\frac{\hbar^{2}}{4m^{2}}\int[\partial^{\alpha}\partial^{\beta}\partial^{\gamma}U(\textbf{r}-\textbf{r}^{\prime})]n_{2}(\textbf{r},\textbf{r}^{\prime},t)d\textbf{r}^{\prime}$
(26)

is the quantum force contribution leading to the quantum fluctuations, and

$F^{\alpha\beta\gamma}=-\int[\partial^{\alpha}U(\textbf{r}-\textbf{r}^{\prime})]\Pi_{2}^{\beta\gamma}(\textbf{r},\textbf{r}^{\prime},t)d\textbf{r}^{\prime}$
(27)

is the interaction contribution containing nonzero limit in the classical
regime, with

$\Pi_{2}^{\alpha\beta}(\textbf{r},\textbf{r}^{\prime},t)=\int dR\sum_{i,j\neq
i}\frac{1}{4m_{i}^{2}}\delta(\textbf{r}-\textbf{r}_{i})\delta(\textbf{r}^{\prime}-\textbf{r}_{j})\times$
$\times[\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi+(\hat{p}_{i}^{\beta})^{*}\Psi^{*}\hat{p}_{i}^{\alpha}\Psi+c.c.].$
(28)

Tensor $\Pi_{2}^{\alpha\beta}(\textbf{r},\textbf{r}^{\prime},t)$ can be
simplified in the correlationless regime to the following form
$\Pi_{2}^{\alpha\beta}(\textbf{r},\textbf{r}^{\prime},t)=\Pi^{\alpha\beta}(\textbf{r},t)\cdot
n(\textbf{r}^{\prime},t)$. However, the correlations caused by the
symmetrization of the bosonic many-particle wave function are used below.

Terms $F^{\alpha\beta\gamma}$ and $F_{qf}^{\alpha\beta\gamma}$ are the third
rank force tensors describing the interparticle interaction. However, equation
(25) contains the flux of tensor $M^{\alpha\beta\gamma}$ which is the fourth
rank tensor appearing in the following form:

$R^{\alpha\beta\gamma\delta}=\int
dR\sum_{i=1}^{N}\delta(\textbf{r}-\textbf{r}_{i})\frac{1}{16m_{i}^{4}}\biggl{[}\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\hat{p}_{i}^{\delta}\Psi$
$+\hat{p}_{i}^{\alpha*}\Psi^{*}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\hat{p}_{i}^{\delta}\Psi+\hat{p}_{i}^{\beta*}\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\gamma}\hat{p}_{i}^{\delta}\Psi+\hat{p}_{i}^{\gamma*}\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi$
$+\hat{p}_{i}^{\delta*}\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi+\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\delta*}\Psi^{*}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\gamma}\Psi$
$+\hat{p}_{i}^{\alpha*}\hat{p}_{i}^{\gamma*}\Psi^{*}\hat{p}_{i}^{\beta}\hat{p}_{i}^{\delta}\Psi+\hat{p}_{i}^{\gamma*}\hat{p}_{i}^{\delta*}\Psi^{*}\hat{p}_{i}^{\alpha}\hat{p}_{i}^{\beta}\Psi+c.c.\biggr{]}.$
(29)

Equation (25) is obtained for bosons with the arbitrary temperature. It can be
separated on two equations for two following subsystems: the BEC and the
normal fluid. All terms in equation (25) are additive on the particles.
Therefore, they are additive on the subsystems. Hence, the structure of the
partial equations is identical to the structure of equation (25):

$\partial_{t}M_{a}^{\alpha\beta\gamma}+\partial_{\delta}R_{a}^{\alpha\beta\gamma\delta}=-\frac{1}{m}n_{a}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}V_{ext}$
$-\frac{1}{m}\Pi_{a}^{\beta\gamma}\partial_{\alpha}V_{ext}-\frac{1}{m}\Pi_{a}^{\alpha\gamma}\partial_{\beta}V_{ext}-\frac{1}{m}\Pi_{a}^{\alpha\beta}\partial_{\gamma}V_{ext}$
$+\frac{1}{m}F_{a,qf}^{\alpha\beta\gamma}+\frac{1}{m}(F_{a}^{\alpha\beta\gamma}+F_{a}^{\beta\gamma\alpha}+F_{a}^{\gamma\alpha\beta}),$
(30)

where subindex $a$ equal $B$ for the BEC and $n$ for the normal fluid.

The fourth rank kinematic tensor $R_{a}^{\alpha\beta\gamma\delta}$ (29) has
the following form after the introduction of the velocity field via the
Madelung transformation of the many-particle wave function:

$R_{a}^{\alpha\beta\gamma\delta}=n_{a}v_{a}^{\alpha}v_{a}^{\beta}v_{a}^{\gamma}v_{a}^{\delta}$
$+v_{a}^{\alpha}v_{a}^{\delta}(p_{a}^{\beta\gamma}+T_{a}^{\beta\gamma})+v_{a}^{\beta}v_{a}^{\delta}(p_{a}^{\alpha\gamma}+T_{a}^{\alpha\gamma})+v_{a}^{\gamma}v_{a}^{\delta}(p_{a}^{\alpha\beta}+T_{a}^{\alpha\beta})$
$+v_{a}^{\alpha}v_{a}^{\gamma}(p_{a}^{\beta\delta}+T_{a}^{\beta\delta})+v_{a}^{\beta}v_{a}^{\gamma}(p_{a}^{\alpha\delta}+T_{a}^{\alpha\delta})+v_{a}^{\alpha}v_{a}^{\beta}(p_{a}^{\gamma\delta}+T_{a}^{\gamma\delta})$
$+v_{a}^{\alpha}Q_{a}^{\beta\gamma\delta}+v_{a}^{\beta}Q_{a}^{\alpha\gamma\delta}+v_{a}^{\gamma}Q_{a}^{\alpha\beta\delta}+v_{a}^{\delta}Q_{a}^{\alpha\beta\gamma}$
$+Q_{a}^{\alpha\beta\gamma\delta}+T_{a}^{\alpha\beta\gamma\delta}+L_{a}^{\alpha\beta\gamma\delta}.$
(31)

This structure shows some similarity to the representations for the second
rank tensor momentum flux (11) and for the third rank tensor (21), where the
higher rank tensors are partially transformed via the concentration, velocity
field and, if possible, via tensors of smaller rank. However, this
transformation is partial since there is the tensor of the equal rank, but
defined in the comoving frame. Moreover, this final tensor is splitted on few
parts. It is two parts for the second rank tensor momentum flux, where we have
the kinetic pressure (quasi-classical part of thermal nature) and the quantum
Bohm potential (the quantum part). There are three parts for the third rank
tensor $M^{\alpha\beta\gamma}$. They are the quasi-classical part of thermal
nature, the quantum part, and the combined thermal-quantum part. For the
fourth rank tensor we also have three parts: the quasi-classical part of
thermal nature $Q_{a}^{\alpha\beta\gamma\delta}$, the quantum part
$T_{a}^{\alpha\beta\gamma\delta}$, and the combined thermal-quantum part
$L_{a}^{\alpha\beta\gamma\delta}$.

Developed model shows that arbitrary quantum system can be modeled via the
hydrodynamic equations which are traditionally associated with the fluid
dynamics. Quantum systems demonstrates that each particle shows the properties
of the wave and this wave-like behavior is incorporated in the quantum
hydrodynamic model. This conclusion follows from the fact that the quantum
hydrodynamic is derived from the Schrodinger equations which contains these
information. These general concept is illustrated for the ultracold bosons,
but the quantum hydrodynamic method can be applied to other physical systems.
This similarity between quantum behavior and the dynamics of fluids recently
found unusual realization. It is experimentally found that classic fluid
objects demonstrate the quantum-like behavior Bush Ch 18 , Couder Nat 05 ,
Bush ARFM 15 . It is observed as the millimetric droplet walking on the
surface of vibrating fluids, where the motion of droplets is affected by the
resonant interaction with their own wave field Couder Nat 05 , Bush ARFM 15 .
Systems walking droplets demonstrate various quantum effects Cristea-Platon Ch
18 , Chowdury Ch 18 , Budanur Ch 19 .

##  III Contribution of interaction in the quantum hydrodynamic equations

Equations (6), (16), and (25) contain terms describing interaction.
Approximate forms of these force fields of different tensor ranks are
necessary to get a truncated set of equations. In our case, it is necessary to
include the short-range nature of the potential of the interparticle
interaction. Moreover, the weak interaction limit is considered. These two
assumptions are used to get simplified form of $F^{\alpha}$,
$F^{\alpha\beta}$, $F^{\alpha\beta\gamma}$ and $F_{qf}^{\alpha\beta\gamma}$ in
this section.

###  III.1 Interaction terms in the Euler equation

The short-range interaction in the Euler for the single species of quantum
particles can be written as the divergence of the symmetric quantum stress
tensor $F^{\alpha}=-\partial^{\beta}\sigma^{\alpha\beta}$.

The first order by the interaction radius approximation gives the following
expression for the quantum stress tensor (see also Andreev PRA08 )

$\sigma^{\alpha\beta}(\textbf{r},t)=-\frac{1}{2}\int dR\sum_{i,j.i\neq
j}\delta(\textbf{r}-\textbf{R}_{ij})\times$
$\times\frac{r^{\alpha}_{ij}r^{\beta}_{ij}}{\mid\textbf{r}_{ij}\mid}\frac{\partial
U(\textbf{r}_{ij})}{\partial\mid\textbf{r}_{ij}\mid}\Psi^{*}(R^{\prime},t)\Psi(R^{\prime},t),$
(32)

where $R^{\prime}=\\{...,\textbf{R}_{ij},...,\textbf{R}_{ij},...\\}$ with
vector $\textbf{R}_{ij}$ located at $i$-th and $j$-th places.

Expression (III.1) can be rewritten in terms of two-particle concentration

$\sigma^{\alpha\beta}(\textbf{r},t)=-\frac{1}{2}Tr(n_{2}(\textbf{r},\textbf{r}^{\prime},t))\int
d\textbf{r}\frac{r^{\alpha}r^{\beta}}{r}\frac{\partial U(r)}{\partial r},$
(33)

where the notion of trace is used

$Trf(\textbf{r},\textbf{r}^{\prime})=f(\textbf{r},\textbf{r}).$ (34)

Consideration of the short-range interaction leads to the separation of
integral containing the potential of interaction. So, the characteristic of
interaction does not depend on the motion or position of particles. This
integral simplifies in the following way

$\int\frac{r^{\alpha}r^{\beta}}{r}\frac{\partial U}{\partial
r}d\textbf{r}=\frac{1}{3}\delta^{\alpha\beta}\int
rU^{\prime}d\textbf{r}=-\delta^{\alpha\beta}\int Ud\textbf{r}.$ (35)

The last integral in this expression is denoted as $g=\int Ud\textbf{r}$.

The two-particle concentration can be calculated in the weakly interacting
limit (see Andreev PRA08 )

$n_{2}(\textbf{r},\textbf{r}^{\prime},t)=n(\textbf{r},t)n(\textbf{r}^{\prime},t)+|\rho(\textbf{r},\textbf{r}^{\prime},t)|^{2}+\wp(\textbf{r},\textbf{r}^{\prime},t),$
(36)

where

$n(\textbf{r},t)=\sum_{f}n_{f}\varphi_{f}^{*}(\textbf{r},t)\varphi_{f}(\textbf{r},t)$
(37)

is the expression of concentration (2) in terms of the single particle wave
functions $\varphi_{f}(\textbf{r},t)$,

$\rho(\textbf{r}^{\prime},\textbf{r},t)=\sum_{f}n_{f}\varphi_{f}^{*}(\textbf{r},t)\varphi_{f}(\textbf{r}^{\prime},t)$
(38)

is the density matrix, and

$\wp(\textbf{r},\textbf{r}^{\prime},t)=\sum_{f}n_{f}(n_{f}-1)|\varphi_{f}(\textbf{r},t)|^{2}|\varphi_{f}(\textbf{r}^{\prime},t)|^{2},$
(39)

The last term in equation describes interaction of pairs of particles being in
the same quantum state. It can be seen from the existence of single quantum
number $g$ in all wave are single-particle wave functions.

Expression (36) can be substituted in the general expression of the force
field (8). However, equation (8) does not contain information about the short-
range nature of considered interaction. The first and second terms are related
to particles located in different quantum states. It cannot be seen from
equation (36), but it follows from intermediate terms which can be found in
Ref. Andreev PRA08 .

The trace of the two-particle concentration entering the quantum stress tensor
has the following form

$Trn_{2}(\textbf{r},\textbf{r}^{\prime},t)\approx
2(n^{2})^{\prime}+n^{2}_{B},$ (40)

where the first term on the right-hand side symbol ’ means that the product of
concentrations is related to the particles in different quantum states.
Therefore, the first term has no $n^{2}_{B}$ contribution from selfaction of
BEC. The dropped terms are described in Ref. Andreev IJMP B 13 .

Present explicit contribution of the BEC concentration $n_{B}$ and the
concentration of normal fluid $n_{n}$ in the first term on the right-hand side
of equation (40):

$(n^{2})^{\prime}=((n_{B}+n_{n})(n_{B}+n_{n}))^{\prime}=(n_{n}^{2}+2n_{B}n_{n}).$
(41)

The last term in equation (40) appears for particles being in BEC. While the
first term on the right-hand side in equation (40) related to interaction of
particles being in different quantum states. Hence, it gives contribution for
the interaction between BEC and normal fluid and for the interaction between
bosons belonging to normal fluid.

Full expression of the quantum stress tensor for the bosons at finite
temperature can be written in terms of the concentration of BEC and the
concentration of normal fluid:

$\sigma^{\alpha\beta}=\frac{1}{2}g\delta^{\alpha\beta}(2n_{n}^{2}+4n_{B}n_{n}+n^{2}_{B}).$
(42)

If we consider dynamics of BEC or normal fluid we cannot use the notion of the
quantum stress tensor $\sigma^{\alpha\beta}$ for the interaction of subspecies
as it is for the interaction of different species.

The first (last) term in equation (42) contains the selfaction of the normal
fluid (of the BEC). The second term in equation (42) presents the interaction
between the BEC and normal fluid.

If we consider dynamics of BEC we need to extract force caused by the BEC and
normal fluid acting on the BEC. This force is the superposition of a part of
the second term in equation (42) and the last term in equation (42):

$F_{B}^{\alpha}=-gn_{B}\partial^{\alpha}(2n_{n}+n_{B}).$ (43)

The second term in equation (42) can be rewritten as follows
$F_{2}^{\alpha}=-2g(n_{B}\partial^{\alpha}n_{n}+n_{n}\partial^{\alpha}n_{B})$.
The first part of this expression is used in equation (43).

If we consider dynamics of normal fluid it means that the source of field in
the first term of $n_{2}$ can be the normal fluid and the BEC, hence the last
term gives no contribution in this case in equation (42):

$F_{n}^{\alpha}=-2gn_{n}\partial^{\alpha}(n_{n}+n_{B}).$ (44)

The nonsymmetric decomposition allows to use the notion of the NLSE. It is
necessary condition to have the GP equation at finite temperatures. Moreover,
the nonsymmetric form is traditionally used in literature Griffin PRB 96 .
Same chose is made at analysis $j_{2}^{\beta}$ below.

####  III.1.1 Nonlinear Schrodinger equations

Dropping the pressure of normal fluid and using the quantum Bohm potential in
form (12) we find a closed set of hydrodynamic equations. Introducing the
macroscopic wave function for both the BEC and the normal fluid for the
potential velocity fields as $\Phi_{a}=\sqrt{n_{a}}e^{\imath
m\phi_{a}/\hbar}$, where $\phi_{a}$ is the potential of the velocity field
$\textbf{v}_{a}=-\nabla\phi_{a}$.

$\imath\hbar\partial_{t}\Phi_{B}=\Biggl{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{ext}+g(n_{B}+2n_{n})\Biggr{)}\Phi_{B},$
(45)

and

$\imath\hbar\partial_{t}\Phi_{n}=\Biggl{(}-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{ext}+2g(n_{B}+n_{n})\Biggr{)}\Phi_{n}.$
(46)

The kinetic energy (the first term on the right-hand side of equations (45)
and (46)) corresponds to the application of the noninteracting limit for the
quantum Bohm potential for the BEC and for the normal fluid.

The pressure of the normal fluid is dropped in equation (46).

Equations (45), (46) correspond to equations 127-129 given in Ref. Dalfovo RMP
99  while there is a difference in the form of presentation.

Therefore, the account of the pressure evolution together with the pressure
flux evolution gives the generalization of the model presented with the
nonlinear Schrodinger equations (45), (46). Necessity of additional equations
is demonstrated in Refs. Andreev 2005 , Andreev 2007 , Andreev 2009  if one
wants to include the quantum fluctuations.

###  III.2 Interaction terms in the pressure evolution equation

General form of the pressure evolution equation contains the interaction via
the force second rank tensor field. Its main contribution is obtained in the
first order by the interaction radius. The result appears in the following
form

$F^{\alpha\beta}(\textbf{r},t)=\frac{1}{8m^{2}}\partial^{\gamma}\int
dR\sum_{i,j;i\neq j}\delta(\textbf{r}-\textbf{R}_{ij})\times$
$\times\frac{r^{\beta}_{ij}r^{\gamma}_{ij}}{\mid\textbf{r}_{ij}\mid}\frac{\partial
U(\textbf{r}_{ij})}{\partial\mid\textbf{r}_{ij}\mid}\biggl{[}\Psi^{*}(R^{\prime},t)(\hat{p}_{(1)}^{\alpha}+\hat{p}_{(2)}^{\alpha})\Psi(R^{\prime},t)+c.c.\biggr{]}$
$-\frac{1}{8m^{2}}\int dR\sum_{i,j;i\neq
j}\delta(\textbf{r}-\textbf{R}_{ij})\frac{r^{\alpha}_{ij}r^{\gamma}_{ij}}{\mid\textbf{r}_{ij}\mid}\frac{\partial
U(\textbf{r}_{ij})}{\partial\mid\textbf{r}_{ij}\mid}\times$
$\times\biggl{[}(\partial^{\gamma}_{(1)}-\partial^{\gamma}_{(2)})\Psi^{*}(R^{\prime},t)(\hat{p}_{(1)}^{\alpha}-\hat{p}_{(2)}^{\alpha})\Psi(R^{\prime},t)$
$+\Psi^{*}(R^{\prime},t)(\partial^{\gamma}_{(1)}-\partial^{\gamma}_{(2)})(\hat{p}_{(1)}^{\alpha}-\hat{p}_{(2)}^{\alpha})\Psi(R^{\prime},t)+c.c.\biggr{]}.$
(47)

Form (47) appears at the expansion of the force tensor field (17) using the
short-range nature of interaction (see Andreev PRA08  for the method described
for the force field, or Andreev 2001  for application of this method to
fermions).

For the force tensor field $F^{\alpha\beta}$ we can present the intermediate
expressions like equations (33) and (36) obtained for the force field
$F^{\alpha}=-\partial^{\beta}\sigma^{\alpha\beta}$. However, similar
expressions obtained for $F^{\alpha\beta}$ are rather large. Hence, we start
the presentation with equation similar to equation (40) obtained after taking
trace of the intermediate expressions.

Therefore, we obtain the following simplification of equation (47) for the
force tensor field $F^{\alpha\beta}$:

$F^{\alpha\beta}=-\frac{g}{4m^{2}}\partial^{\beta}[2(n\Lambda^{\alpha})^{\prime}+n_{B}\Lambda^{\alpha}_{B}]$
$+\frac{\imath}{\hbar}\frac{g}{4m^{2}}[2(nr^{\alpha\beta})^{\prime}-2(\Lambda^{\alpha}\Lambda^{\beta})^{\prime}+n_{B}r^{\alpha\beta}_{B}-\Lambda^{\alpha}_{B}\Lambda^{\beta}_{B}]+c.c.,$
(48)

where we use the intermediate functions $\Lambda^{\alpha}$ and
$r^{\alpha\beta}$ with the following definitions:

$\Lambda^{\alpha}=\sum_{f}n_{f}\varphi_{f}^{*}\hat{p}^{\alpha}\varphi_{f}=mj^{\alpha}-\imath\frac{\hbar}{2}\partial^{\alpha}n,$
(49)

and

$r^{\alpha\beta}=\sum_{f}n_{f}\varphi_{f}^{*}\hat{p}^{\alpha}\hat{p}^{\beta}\varphi_{f}$
$=m^{2}\biggl{(}nv^{\alpha}v^{\beta}+p^{\alpha\beta}-\frac{\hbar^{2}}{m^{2}}\sum_{f}n_{f}a_{f}\partial^{\alpha}\partial^{\beta}a_{f}\biggr{)}$
$-\imath\frac{m\hbar}{2}[\partial^{\alpha}(nv^{\beta})+\partial^{\beta}(nv^{\alpha})].$
(50)

The calculation of functions $\Lambda^{\alpha}$ and $r^{\alpha\beta}$ includes
the Madelung transformation of the single-particle wave functions
$\varphi_{f}(r,t)=\sqrt{a_{f}}e^{\imath S_{f}}$. Next, we use the following
definitions of the velocity field and the pressure tensor in terms of the
single-particle wave functions
$nv^{\alpha}=\sum_{f}n_{f}a_{f}^{2}(\hbar\partial^{\alpha}S_{f}/m)$, and
$p^{\alpha\beta}=\sum_{f}n_{f}a_{f}^{2}u_{f}^{\alpha}u_{f}^{\beta}$, where
$u_{f}^{\alpha}=(\hbar\partial^{\alpha}S_{f}/m)-v^{\alpha}$.

Let us represent terms like $(n\Lambda^{\alpha})^{\prime}$ in the explicit
form:

$F^{\alpha\beta}=-\frac{g}{4m^{2}}\partial^{\beta}[2n_{n}\Lambda^{\alpha}_{n}$
$+2n_{n}\Lambda^{\alpha}_{B}+2n_{B}\Lambda^{\alpha}_{n}+n_{B}\Lambda^{\alpha}_{B}]$
$+\frac{\imath}{\hbar}\frac{g}{4m^{2}}[2n_{n}r^{\alpha\beta}_{n}-2\Lambda^{\alpha}_{n}\Lambda^{\beta}_{n}+2n_{n}r^{\alpha\beta}_{B}-2\Lambda^{\alpha}_{n}\Lambda^{\beta}_{B}$
$2n_{B}r^{\alpha\beta}_{n}-2\Lambda^{\alpha}_{B}\Lambda^{\beta}_{n}+n_{B}r^{\alpha\beta}_{B}-\Lambda^{\alpha}_{B}\Lambda^{\beta}_{B}]+c.c..$
(51)

Further calculation gives the representation of tensor $F^{\alpha\beta}$ in
term of hydrodynamic functions:

$F^{\alpha\beta}=-\frac{g}{2m}\partial^{\beta}[2n_{n}j_{n}^{\alpha}+2n_{n}j_{B}^{\alpha}+2n_{B}j_{n}^{\alpha}+n_{B}j_{B}^{\alpha}]$
$+\frac{g}{4m}\biggl{[}2n_{n}(\partial j_{n}+\partial j_{n})-2j_{n}\partial
n_{n}-2j_{n}\partial n_{n}$
$+2n_{n}(\partial^{\alpha}j_{B}^{\beta}+\partial^{\beta}j_{B}^{\alpha})-2j_{B}^{\alpha}\partial^{\beta}n_{n}-2j_{B}^{\beta}\partial^{\alpha}n_{n}$
$+2n_{B}(\partial^{\alpha}j_{n}^{\beta}+\partial^{\beta}j_{n}^{\alpha})-2j_{n}^{\alpha}\partial^{\beta}n_{B}-2j_{n}^{\beta}\partial^{\alpha}n_{B}$
$+n_{B}(\partial^{\alpha}j_{B}^{\beta}+\partial^{\beta}j_{B}^{\alpha})-j_{B}^{\alpha}\partial^{\beta}n_{B}-j_{B}^{\beta}\partial^{\alpha}n_{B}\biggr{]}.$
(52)

The momentum flux evolution equation contains the symmetric combination of the
force tensor fields $F^{\alpha\beta}$:

$F^{\alpha\beta}+F^{\beta\alpha}=-\frac{g}{m}\biggl{[}2(j_{n}^{\alpha}\partial^{\beta}n_{n}+j_{n}^{\beta}\partial^{\alpha}n_{n})$
$+2(j_{n}^{\alpha}\partial^{\beta}n_{B}+j_{n}^{\beta}\partial^{\alpha}n_{B})$
$+2(j_{B}^{\alpha}\partial^{\beta}n_{n}+j_{B}^{\beta}\partial^{\alpha}n_{n})+j_{B}^{\alpha}\partial^{\beta}n_{B}+j_{B}^{\beta}\partial^{\alpha}n_{B}\biggr{]}.$
(53)

The zero temperature analysis demonstrates that there is nonzero pressure for
the BECs, caused by the quantum fluctuations entering the set of hydrodynamic
equations via the evolution of the pressure flux Andreev 2005 , Andreev 2007 ,
Andreev 2009 . The pressure also exists for the normal fluid. So, we make
decomposition of the momentum flux evolution equation on two partial equations
for $\Pi^{\alpha\beta}_{n}$ and $\Pi^{\alpha\beta}_{B}$. Formally this
decomposition is presented with equation (20). To complete this procedure we
need to split the force tensor field $F^{\alpha\beta}+F^{\beta\alpha}$
$=F_{B}^{\alpha\beta}+F_{B}^{\beta\alpha}$
$+F_{n}^{\alpha\beta}+F_{n}^{\beta\alpha}$, where

$F_{B}^{\alpha\beta}+F_{B}^{\beta\alpha}=-\frac{g}{m}\biggl{[}2(j_{B}^{\alpha}\partial^{\beta}n_{n}+j_{B}^{\beta}\partial^{\alpha}n_{n})$
$+j_{B}^{\alpha}\partial^{\beta}n_{B}+j_{B}^{\beta}\partial^{\alpha}n_{B}\biggr{]},$
(54)

and

$F_{n}^{\alpha\beta}+F_{n}^{\beta\alpha}=-\frac{g}{m}\biggl{[}2(j_{n}^{\alpha}\partial^{\beta}n_{n}+j_{n}^{\beta}\partial^{\alpha}n_{n})$
$+2(j_{n}^{\alpha}\partial^{\beta}n_{B}+j_{n}^{\beta}\partial^{\alpha}n_{B})\biggr{]}.$
(55)

After extraction of the pressure tensor $p^{\alpha\beta}$ from the momentum
flux evolution $\Pi^{\alpha\beta}$ we have extra contribution of the
interaction in the pressure evolution equation in compare with equations (20).
It contains the following contribution $F^{\alpha\beta}-v^{\beta}F^{\alpha}$.

Using equations (43), (44), (55), (54) find
$F^{\alpha\beta}+F^{\alpha\beta}-v^{\alpha}F^{\beta}-v^{\beta}F^{\alpha}=0$
for the BECs and for the normal fluid.

A pressure evolution equation is used in Kavoulakis PRA 98  for bosons above
the critical temperature. Equation 4 of Ref. Kavoulakis PRA 98  contains the
force in the following form $n\textbf{v}\cdot\textbf{F}$ which generally
differs from $F^{\alpha\beta}-v^{\beta}F^{\alpha}$ obtained above.

###  III.3 The short-range interaction in the third rank tensor evolution
equation

The third rank tensor $M^{\alpha\beta\gamma}$ (25) evolution equation contains
two kinds of the third rank force tensors $F^{\alpha\beta\gamma}$ (27) and
$F_{qf}^{\alpha\beta\gamma}$ (26). Consider them separately.

####  III.3.1 Quasi-classical third rank force tensor

Tensor $F_{qf}^{\alpha\beta\gamma}$ (26) is proportional to the Planck
constant, so it goes to zero in the classical limit. The third rank force
tensor $F^{\alpha\beta\gamma}$ (27) is different, it has nonzero limit in the
quasiclassical regime. However, we are interested in the value of tensor
$F^{\alpha\beta\gamma}$ (27) in one of quantum regimes for the degenerate
bosons.

We calculate the third rank force tensor $F^{\alpha\beta\gamma}$ (27) in the
first order by the interaction radius appears in the following form

$F^{\alpha\beta\gamma}(\textbf{r},t)=\frac{1}{8m^{3}}\partial^{\mu}\int
dR\sum_{i,j;i\neq
j}\delta(\textbf{r}-\textbf{R}_{ij})\frac{r^{\mu}_{ij}r^{\gamma}_{ij}}{\mid\textbf{r}_{ij}\mid}\frac{\partial
U(\textbf{r}_{ij})}{\partial\mid\textbf{r}_{ij}\mid}\biggl{[}\Psi^{*}(R^{\prime},t)\hat{p}_{(1)}^{\alpha}\hat{p}_{(1)}^{\beta}\Psi(R^{\prime},t)+\hat{p}_{(1)}^{\beta*}\Psi^{*}(R^{\prime},t)\hat{p}_{(1)}^{\alpha}\Psi(R^{\prime},t)+c.c.\biggr{]}$

$-\frac{1}{8m^{3}}\int dR\sum_{i,j;i\neq
j}\delta(\textbf{r}-\textbf{R}_{ij})\frac{r^{\mu}_{ij}r^{\gamma}_{ij}}{\mid\textbf{r}_{ij}\mid}\frac{\partial
U(\textbf{r}_{ij})}{\partial\mid\textbf{r}_{ij}\mid}\biggl{[}(\partial^{\mu}_{(1)}-\partial^{\mu}_{(2)})\Psi^{*}(R^{\prime},t)\hat{p}_{(1)}^{\alpha}\hat{p}_{(1)}^{\beta}\Psi(R^{\prime},t)$
$+\Psi^{*}(R^{\prime},t)(\partial^{\mu}_{(1)}-\partial^{\mu}_{(2)})\hat{p}_{(1)}^{\alpha}\hat{p}_{(1)}^{\beta}\Psi(R^{\prime},t)+\hat{p}_{(1)}^{\beta*}(\partial^{\mu}_{(1)}-\partial^{\mu}_{(2)})\Psi^{*}(R^{\prime},t)\hat{p}_{(1)}^{\alpha}\Psi(R^{\prime},t)+\hat{p}_{(1)}^{\beta*}\Psi^{*}(R^{\prime},t)(\partial^{\mu}_{(1)}-\partial^{\mu}_{(2)})\hat{p}_{(1)}^{\alpha}\Psi(R^{\prime},t)+c.c.\biggr{]}.$
(56)

Here, the part of expression for $F^{\alpha\beta\gamma}$ containing the
interaction potential appears as the independent multiplier. It has same form
as the integral in the Euler equation (35). Hence, tensor
$F^{\alpha\beta\gamma}$ is proportional to the Groos-Pitaevskii interaction
constant.

Further calculation in the weakly interacting limit, following the method
described in Ref. Andreev PRA08 , gives an intermediate representation of the
third rank force tensor:

$F^{\alpha\beta\gamma}(\textbf{r},t)=-\frac{g}{4m^{3}}\Biggl{[}\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{B}+\Pi^{\alpha\beta}\partial^{\gamma}n+\biggl{(}n\sum_{f}n_{f}\partial^{\gamma}\varphi_{f}^{*}\hat{p}^{\alpha}\hat{p}^{\beta}\varphi_{f}+\frac{\imath}{\hbar}\Lambda^{\gamma}r^{\alpha\beta}-\frac{\imath}{\hbar}\Lambda^{\alpha*}\kappa^{\gamma\beta}+\frac{\imath}{\hbar}\Lambda^{\beta}\kappa^{\alpha\gamma}+c.c.\biggr{)}\Biggr{]},$
(57)

where

$\kappa^{\alpha\beta}=\sum_{f}n_{f}p^{\alpha}\varphi_{f}^{*}\cdot\hat{p}^{\beta}\varphi_{f}.$
(58)

Function $\kappa^{\alpha\beta}$ is the nonsymmetric tensor. It has symmetric
real part and the antisymmetric imaginary part:

$\kappa^{\alpha\beta}=m^{2}\biggl{(}nv^{\alpha}v^{\beta}+p^{\alpha\beta}+\frac{\hbar^{2}}{m^{2}}\sum_{f}n_{f}\partial^{\alpha}\cdot
a_{f}\partial^{\beta}a_{f}\biggr{)}$ $-\frac{1}{2}\imath
m\hbar[v^{\alpha}\partial^{\beta}n-v^{\beta}\partial^{\alpha}n+\sum_{f}n_{f}a_{f}(u^{\alpha}\partial^{\beta}a_{f}-u^{\beta}\partial^{\alpha}a_{f})].$
(59)

No specific notation is introduced for the third rank tensor
$\sum_{f}n_{f}\partial^{\gamma}\varphi_{f}^{*}\hat{p}^{\alpha}\hat{p}^{\beta}\varphi_{f}$.
In our calculations we need its imaginary part multiplied by 2:

$\sum_{f}n_{f}\partial^{\gamma}\varphi_{f}^{*}\hat{p}^{\alpha}\hat{p}^{\beta}\varphi_{f}+c.c.=2m^{2}\Biggl{[}\frac{1}{2}\partial^{\gamma}n\cdot
v^{\alpha}v^{\beta}-\partial^{\alpha}n\cdot
v^{\beta}v^{\gamma}-\partial^{\beta}n\cdot
v^{\alpha}v^{\gamma}-nv^{\gamma}(\partial^{\beta}v^{\alpha}+\partial^{\alpha}v^{\beta})$
$+\sum_{f}n_{f}a_{f}(\partial^{\gamma}a_{f})u_{f}^{\alpha}u_{f}^{\beta}-\frac{1}{2}\partial^{\beta}p^{\alpha\gamma}-\frac{1}{2}\partial^{\alpha}p^{\beta\gamma}+\frac{1}{2}\sum_{f}n_{f}a_{f}^{2}(u_{f}^{\beta}\partial^{\alpha}u_{f}^{\gamma}+u_{f}^{\alpha}\partial^{\beta}u_{f}^{\gamma})$
$+v^{\alpha}\sum_{f}n_{f}a_{f}(\partial^{\gamma}a_{f}\cdot
u_{f}^{\beta}-\partial^{\beta}a_{f}\cdot
u_{f}^{\gamma})+v^{\beta}\sum_{f}n_{f}a_{f}(\partial^{\gamma}a_{f}\cdot
u_{f}^{\alpha}-\partial^{\alpha}a_{f}\cdot
u_{f}^{\gamma})-\frac{\hbar^{2}}{m^{2}}\sum_{f}n_{f}\partial^{\gamma}a_{f}\cdot\partial^{\alpha}\partial^{\beta}a_{f}\biggr{)}\Biggr{]}.$
(60)

Equation (57) includes term $\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{B}$
which describes full contribution of the BEC in $F^{\alpha\beta\gamma}$.

The second term in equation (57) is an analog of the first term in equation
(36). All following terms in equation (57) are the analog of the second term
in equation (36). It can be interpreted as the exchange interaction.

Further calculation of the (57) gives the partially truncated expression
mainly presented via the macroscopic hydrodynamic functions:

$F^{\alpha\beta\gamma}(\textbf{r},t)=-\frac{g}{4}\biggl{[}4\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{B}+4\Pi^{\alpha\beta}\partial^{\gamma}n+\partial^{\alpha}n\biggl{(}\Pi^{\beta\gamma}+\frac{\hbar^{2}}{4m^{2}}\partial^{\beta}\partial^{\gamma}n\biggr{)}+\partial^{\beta}n\biggl{(}\Pi^{\alpha\gamma}+\frac{\hbar^{2}}{4m^{2}}\partial^{\alpha}\partial^{\gamma}n\biggr{)}+\partial^{\gamma}n\biggl{(}\Pi^{\alpha\beta}-\frac{\hbar^{2}}{4m^{2}}\partial^{\alpha}\partial^{\beta}n\biggr{)}$
$+n\biggl{(}3\partial^{\gamma}n\cdot
v^{\alpha}v^{\beta}-2\partial^{\alpha}n\cdot
v^{\beta}v^{\gamma}-2\partial^{\beta}n\cdot
v^{\alpha}v^{\gamma}-nv^{\gamma}(\partial^{\alpha}v^{\beta}+\partial^{\beta}v^{\alpha})-\partial^{\beta}p^{\alpha\gamma}-\partial^{\alpha}p^{\beta\gamma}$
$+\sum_{f}n_{f}(\partial^{\gamma}a_{f}^{2})u_{f}^{\alpha}u_{f}^{\beta}+\sum_{f}n_{f}a_{f}^{2}(u_{f}^{\beta}\partial^{\alpha}u_{f}^{\gamma}+u_{f}^{\alpha}\partial^{\beta}u_{f}^{\gamma})+\frac{3}{2}v^{\alpha}\sum_{f}n_{f}(\partial^{\gamma}a_{f}^{2}\cdot
u_{f}^{\beta}-\partial^{\beta}a_{f}^{2}\cdot u_{f}^{\gamma})$
$+\frac{3}{2}v^{\beta}\sum_{f}n_{f}(\partial^{\gamma}a_{f}^{2}\cdot
u_{f}^{\alpha}-\partial^{\alpha}a_{f}^{2}\cdot
u_{f}^{\gamma})-2\frac{\hbar^{2}}{m^{2}}\sum_{f}n_{f}\partial^{\gamma}a_{f}\cdot\partial^{\alpha}\partial^{\beta}a_{f}\biggr{)}\biggr{]}.$
(61)

Equation for the third rank tensor $M^{\alpha\beta\gamma}$ evolution contains
the symmetric combination of the third rank force tensors (61) which cam be
presented in the following form:

$F^{\alpha\beta\gamma}+F^{\beta\gamma\alpha}+F^{\gamma\alpha\beta}=-\frac{g}{4}\biggl{[}4(\Pi_{B}^{\beta\gamma}\partial^{\alpha}n_{B}+\Pi_{B}^{\alpha\gamma}\partial^{\beta}n_{B}+\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{B})+4(\Pi^{\beta\gamma}\partial^{\alpha}n+\Pi^{\alpha\gamma}\partial^{\beta}n+\Pi^{\alpha\beta}\partial^{\gamma}n)$
$+\partial^{\alpha}n\biggl{(}3\Pi^{\beta\gamma}+\frac{\hbar^{2}}{4m^{2}}\partial^{\beta}\partial^{\gamma}n\biggr{)}+\partial^{\beta}n\biggl{(}3\Pi^{\alpha\gamma}+\frac{\hbar^{2}}{4m^{2}}\partial^{\alpha}\partial^{\gamma}n\biggr{)}+\partial^{\gamma}n\biggl{(}3\Pi^{\alpha\beta}+\frac{\hbar^{2}}{4m^{2}}\partial^{\alpha}\partial^{\beta}n\biggr{)}$
$-n(\partial^{\alpha}\Pi^{\beta\gamma}+\partial^{\beta}\Pi^{\alpha\gamma}+\partial^{\gamma}\Pi^{\alpha\beta})-\frac{3\hbar^{2}}{4m^{2}}n\partial^{\alpha}\partial^{\beta}\partial^{\gamma}n\biggr{]}.$
(62)

The pressure flux evolution equation obtained as the reduction of the third
rank tensor $M^{\alpha\beta\gamma}$ evolution equation contains the following
combination of the force fields:

$F^{\alpha\beta\gamma}+F^{\beta\gamma\alpha}+F^{\gamma\alpha\beta}-\frac{1}{mn}(F^{\alpha}\Pi^{\beta\gamma}+F^{\beta}\Pi^{\alpha\gamma}+F^{\gamma}\Pi^{\alpha\beta})=\frac{g}{4}[\partial^{\alpha}(n\Pi^{\beta\gamma})+\partial^{\beta}(n\Pi^{\alpha\gamma})+\partial^{\gamma}(n\Pi^{\alpha\beta})]^{\prime}$
$+\frac{g\hbar^{2}}{16m^{2}}[3n\partial^{\alpha}\partial^{\beta}\partial^{\gamma}n-\partial^{\alpha}n\cdot\partial^{\beta}\partial^{\gamma}n-\partial^{\beta}n\cdot\partial^{\alpha}\partial^{\gamma}n-\partial^{\gamma}n\cdot\partial^{\alpha}\partial^{\beta}n]^{\prime},$
(63)

where symbol $[]^{\prime}$ specifies that product of functions describing the
BEC is excluded similarly equations (40) and (41).

The quantum part can be represented

$F^{\alpha\beta\gamma}+F^{\beta\gamma\alpha}+F^{\gamma\alpha\beta}-\frac{1}{mn}(F^{\alpha}\Pi^{\beta\gamma}+F^{\beta}\Pi^{\alpha\gamma}+F^{\gamma}\Pi^{\alpha\beta})=\frac{g}{4}[\partial^{\alpha}(n\Pi^{\beta\gamma})+\partial^{\beta}(n\Pi^{\alpha\gamma})+\partial^{\gamma}(n\Pi^{\alpha\beta})]^{\prime}$
$+\frac{g\hbar^{2}}{16m^{2}}[\partial^{\alpha}(n\partial^{\beta}\partial^{\gamma}n-\partial^{\beta}n\cdot\partial^{\gamma}n)+\partial^{\beta}(n\partial^{\alpha}\partial^{\gamma}n-\partial^{\alpha}n\cdot\partial^{\gamma}n)+\partial^{\gamma}(n\partial^{\alpha}\partial^{\beta}n-\partial^{\alpha}n\cdot\partial^{\beta}n)]^{\prime}.$
(64)

It can be considered as
$F^{\alpha\beta\gamma}+F^{\beta\gamma\alpha}+F^{\gamma\alpha\beta}-\frac{1}{mn}(F^{\alpha}\Pi^{\beta\gamma}+F^{\beta}\Pi^{\alpha\gamma}+F^{\gamma}\Pi^{\alpha\beta})$
$=\tilde{F}^{\alpha\beta\gamma}+\tilde{F}^{\beta\gamma\alpha}+\tilde{F}^{\gamma\alpha\beta}$,
where

$\tilde{F}^{\alpha\beta\gamma}=\frac{g}{4}[\partial^{\alpha}(n\Pi^{\beta\gamma})+\frac{g\hbar^{2}}{16m^{2}}[\partial^{\alpha}(n\partial^{\beta}\partial^{\gamma}n-\partial^{\beta}n\cdot\partial^{\gamma}n)]^{\prime}.$
(65)

It is the derivative of the second rank tensor.

If we consider the zero temperature limit we find
$F^{\alpha\beta\gamma}=(-g/4m^{3})\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{B}$.
The transition from the equation of evolution for tensor
$M^{\alpha\beta\gamma}$ to the equation of evolution for tensor
$Q^{\alpha\beta\gamma}$, which is the sibling of $M^{\alpha\beta\gamma}$, but
the pressure flux $Q^{\alpha\beta\gamma}$ is defined in the comoving frame
leads to the canceling of such term. Hence, the nonzero contribution comes
from the quantum part of the third rank force tensor
$F_{qf}^{\alpha\beta\gamma}$. The nonzero temperature gives a nonzero
contribution of $F^{\alpha\beta\gamma}$ at the transition to the pressure flux
evolution equation.

Equation (65) is obtained for all bosons. We need to separate it on the force
acting on the BEC and the force acting on the normal fluid.

Finally, we obtain

$\tilde{F}_{n}^{\alpha\beta\gamma}+\tilde{F}_{n}^{\beta\gamma\alpha}+\tilde{F}_{n}^{\gamma\alpha\beta}=\frac{g}{4}[\partial^{\alpha}(n_{n}\Pi_{n}^{\beta\gamma})+\partial^{\beta}(n_{n}\Pi_{n}^{\alpha\gamma})+\partial^{\gamma}(n_{n}\Pi_{n}^{\alpha\beta})$
$+n_{n}\partial^{\alpha}(\Pi_{B}^{\beta\gamma}+\partial^{\beta}\Pi_{B}^{\alpha\gamma}+\partial^{\gamma}\Pi_{B}^{\alpha\beta})+\Pi_{n}^{\beta\gamma}\partial^{\alpha}n_{B}+\Pi_{n}^{\alpha\gamma}\partial^{\beta}n_{B}+\Pi_{n}^{\alpha\beta}\partial^{\gamma}n_{B}]$
$+\frac{g\hbar^{2}}{16m^{2}}\biggl{[}3n_{n}\partial^{\alpha}\partial^{\beta}\partial^{\gamma}n_{n}+3n_{n}\partial^{\alpha}\partial^{\beta}\partial^{\gamma}n_{B}-\partial^{\alpha}n_{n}\cdot\partial^{\beta}\partial^{\gamma}n_{n}-\partial^{\beta}n_{n}\cdot\partial^{\alpha}\partial^{\gamma}n_{n}-\partial^{\gamma}n_{n}\cdot\partial^{\alpha}\partial^{\beta}n_{n}$
$-\frac{1}{2}\partial^{\alpha}n_{n}\cdot\partial^{\beta}\partial^{\gamma}n_{B}-\frac{1}{2}\partial^{\beta}n_{n}\cdot\partial^{\alpha}\partial^{\gamma}n_{B}-\frac{1}{2}\partial^{\gamma}n_{n}\cdot\partial^{\alpha}\partial^{\beta}n_{B}-\frac{1}{2}\partial^{\alpha}n_{B}\cdot\partial^{\beta}\partial^{\gamma}n_{n}-\frac{1}{2}\partial^{\beta}n_{B}\cdot\partial^{\alpha}\partial^{\gamma}n_{n}-\frac{1}{2}\partial^{\gamma}n_{B}\cdot\partial^{\alpha}\partial^{\beta}n_{n}\biggr{]},$
(66)

and

$\tilde{F}_{B}^{\alpha\beta\gamma}+\tilde{F}_{B}^{\beta\gamma\alpha}+\tilde{F}_{B}^{\gamma\alpha\beta}=\frac{g}{4}[n_{B}\partial^{\alpha}(\Pi_{n}^{\beta\gamma}+\partial^{\beta}\Pi_{n}^{\alpha\gamma}+\partial^{\gamma}\Pi_{n}^{\alpha\beta})$
$+\Pi_{B}^{\beta\gamma}\partial^{\alpha}n_{n}+\Pi_{B}^{\alpha\gamma}\partial^{\beta}n_{n}+\Pi_{B}^{\alpha\beta}\partial^{\gamma}n_{n}]+\frac{g\hbar^{2}}{16m^{2}}\biggl{[}3n_{B}\partial^{\alpha}\partial^{\beta}\partial^{\gamma}n_{n}$
$-\frac{1}{2}\partial^{\alpha}n_{B}\cdot\partial^{\beta}\partial^{\gamma}n_{n}-\frac{1}{2}\partial^{\beta}n_{B}\cdot\partial^{\alpha}\partial^{\gamma}n_{n}-\frac{1}{2}\partial^{\gamma}n_{B}\cdot\partial^{\alpha}\partial^{\beta}n_{n}-\frac{1}{2}\partial^{\alpha}n_{n}\cdot\partial^{\beta}\partial^{\gamma}n_{B}-\frac{1}{2}\partial^{\beta}n_{n}\cdot\partial^{\alpha}\partial^{\gamma}n_{B}-\frac{1}{2}\partial^{\gamma}n_{n}\cdot\partial^{\alpha}\partial^{\beta}n_{B}\biggr{]}.$
(67)

Equations (66) and (67) gives final expressions for the quasi-classic force
fields in the pressure flux evolution equations for two fluid model.

####  III.3.2 The third rank force tensor describing the quantum fluctuation

The quantum fluctuations in the zero temperature BECs is caused by tensor
$F_{qf}^{\alpha\beta\gamma}$ (26). Its major contribution can be found in the
first order by the interaction radius approximation. Here, we consider the
small nonzero temperature regime of $F_{qf}^{\alpha\beta\gamma}$ for the
bosons. So, we obtain its generalization for the two-fluid model. The quantum
third rank force tensor $F_{qf}^{\alpha\beta\gamma}$ (26) is calculated in the
first order by the interaction radius

$F_{qf}^{\alpha\beta\gamma}=-\frac{\hbar^{2}}{8m^{2}}\partial^{\delta}\int
dR\sum_{i,j.i\neq j}\delta(\textbf{r}-\textbf{R}_{ij})\times$ $\times
r^{\delta}_{ij}\partial^{\alpha}_{i}\partial^{\beta}_{i}\partial^{\gamma}_{i}U(\textbf{r}_{ij})\Psi^{*}(R^{\prime},t)\Psi(R^{\prime},t).$
(68)

In formula (68) for tensor $F_{qf}^{\alpha\beta\gamma}$ we have separation of
the integral containing the interaction potential, as we have at the
calculation of other force fields above. However, here we obtain different
integral $\int r^{\alpha}\partial^{\beta}\partial^{\gamma}\partial^{\delta}$.
Calculation of this integral leads to the second interaction constant given
below in the simplified expression for the quantum third rank force tensor

$F_{qf}^{\alpha\beta\gamma}=\frac{\hbar^{2}}{8m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}\partial^{\delta}Trn_{2}(\textbf{r},\textbf{r}^{\prime},t),$
(69)

where

$g_{2}=\frac{2}{3}\int d\textbf{r}U^{\prime\prime}(r),$ (70)

and

$I_{0}^{\alpha\beta\gamma\delta}=\delta^{\alpha\beta}\delta^{\gamma\delta}+\delta^{\alpha\gamma}\delta^{\beta\delta}+\delta^{\alpha\delta}\delta^{\beta\gamma}.$
(71)

Calculation of the two-particle concentration leads to the following
expression:

$F_{qf}^{\alpha\beta\gamma}=\frac{\hbar^{2}}{8m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}\partial^{\delta}(2n_{n}^{2}+4n_{B}n_{n}+n_{B}^{2}).$
(72)

Here we have $\partial^{\delta}(2n_{n}^{2}+4n_{B}n_{n}+n_{B}^{2})$. Next, we
open brackets and find
$(4n_{n}\partial^{\delta}n_{n}+4n_{n}\partial^{\delta}n_{B}+4n_{B}\partial^{\delta}n_{n}+2n_{B}\partial^{\delta}n_{B})$.
The source of field is under derivative, while the multiplier in front
corresponds to the species under the action of field. Hence the first two
terms correspond to the force acting on the normal fluid, while the third and
fourth terms correspond to the force acting on the BEC.

Therefore, we can separate expression (72) on two parts corresponding to the
BEC and to the normal fluid:

$F_{qf,B}^{\alpha\beta\gamma}=\frac{\hbar^{2}}{4m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}(2n_{B}\partial^{\delta}n_{n}+n_{B}\partial^{\delta}n_{B}),$
(73)

and

$F_{qf,n}^{\alpha\beta\gamma}=\frac{\hbar^{2}}{2m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}\partial^{\delta}(n_{n}\partial^{\delta}n_{n}+n_{B}\partial^{\delta}n_{n}).$
(74)

##  IV Hydrodynamic equations for two fluid model of bosons with nonzero
temperature

This section provides the final set of equations obtained in this paper. The
method of the introduction of the velocity field and corresponding
representation of the hydrodynamic equations is not described in this paper.
It can be found in number other papers, majority of details are given in Refs.
Andreev PRA08 , Andreev 2001 .

In this regime we have two continuity equations:

$\partial_{t}n_{B}+\nabla\cdot(n_{B}\textbf{v}_{B})=0,$ (75)

and

$\partial_{t}n_{n}+\nabla\cdot(n_{n}\textbf{v}_{n})=0.$ (76)

The Euler equation for bosons in the BEC state

$mn_{B}(\partial_{t}+\textbf{v}_{B}\cdot\nabla)v^{\alpha}_{B}+\partial_{\beta}T_{B}^{\alpha\beta}+\partial_{\beta}p_{qf}^{\alpha\beta}$
$+gn_{B}\partial^{\alpha}n_{B}=-n_{B}\partial^{\alpha}V_{ext}-2gn_{B}\partial^{\alpha}n_{n},$
(77)

where the quantum Bohm potential is given by equation (12) in the
noninteracting limit.

The Euler equation for bosons in the excited states corresponding to the
nonzero temperature

$mn_{n}(\partial_{t}+\textbf{v}_{n}\cdot\nabla)v^{\alpha}_{n}+\partial_{\beta}p_{n,eff}^{\alpha\beta}$
$+2gn_{n}\partial^{\alpha}n_{n}=-n_{n}\partial^{\alpha}V_{ext}-2gn_{n}\partial^{\alpha}n_{B},$
(78)

where the effective pressure tensor
$p_{n,eff}^{\alpha\beta}=p_{n}^{\alpha\beta}+T_{n}^{\alpha\beta}$.

The effective pressure evolution equation for normal boson fluid is also a
part of developed and applied hydrodynamic model

$\partial_{t}p_{n,eff}^{\alpha\beta}+v_{n}^{\gamma}\partial_{\gamma}p_{n,eff}^{\alpha\beta}+p_{n,eff}^{\alpha\gamma}\partial_{\gamma}v_{n}^{\beta}+p_{n,eff}^{\beta\gamma}\partial_{\gamma}v_{n}^{\alpha}$
$+p_{n,eff}^{\alpha\beta}\partial_{\gamma}v_{n}^{\gamma}+\partial_{\gamma}T^{\alpha\beta\gamma}_{n}+\partial_{\gamma}Q^{\alpha\beta\gamma}_{n}=0.$
(79)

Moreover, we have the pressure (the quantum Bohm potential)
$p_{B,eff}^{\alpha\beta}=T_{B}^{\alpha\beta}+p_{qf}^{\alpha\beta}$ evolution
equation for the BEC

$\partial_{t}p_{B,eff}^{\alpha\beta}+v_{B}^{\gamma}\partial_{\gamma}p_{B,eff}^{\alpha\beta}+p_{B,eff}^{\alpha\gamma}\partial_{\gamma}v_{B}^{\beta}+p_{B,eff}^{\beta\gamma}\partial_{\gamma}v_{B}^{\alpha}$
$+p_{B,eff}^{\alpha\beta}\partial_{\gamma}v_{B}^{\gamma}+\partial_{\gamma}T^{\alpha\beta\gamma}_{B}+\partial_{\gamma}Q^{\alpha\beta\gamma}_{qf}=0.$
(80)

Let us to point out the following property of the quantum Bohm potential that
it satisfies the following equation for the arbitrary species $a$

$\partial_{t}T_{a}^{\alpha\beta}+v_{a}^{\gamma}\partial_{\gamma}T_{a}^{\alpha\beta}+T_{a}^{\alpha\gamma}\partial_{\gamma}v_{a}^{\beta}+T_{a}^{\beta\gamma}\partial_{\gamma}v_{a}^{\alpha}$
$+T_{a}^{\alpha\beta}\partial_{\gamma}v_{a}^{\gamma}+\partial_{\gamma}T^{\alpha\beta\gamma}_{a}=0.$
(81)

It is expected that approximate form of the quantum Bohm potential (12)
satisfies equation (81) existing at the zero interaction. Hence, substitute
(12) in equation (81) with the zero right-hand side:

$\partial^{\beta}\partial^{\gamma}n_{a}\cdot(\partial^{\gamma}v_{a}^{\alpha}-\partial^{\alpha}v_{a}^{\gamma})+\partial^{\alpha}\partial^{\gamma}n_{a}\cdot(\partial^{\gamma}v_{a}^{\beta}-\partial^{\beta}v_{a}^{\gamma})$
$+\frac{1}{3}\partial_{\gamma}n_{a}\cdot(\partial^{\beta}\partial^{\gamma}v_{a}^{\alpha}+\partial^{\alpha}\partial^{\gamma}v_{a}^{\beta}-\partial^{\alpha}\partial^{\beta}v_{a}^{\gamma})$
$+\frac{1}{3}n_{a}[\triangle(\partial^{\beta}v_{a}^{\alpha}+\partial^{\alpha}v_{a}^{\beta})-\partial^{\alpha}\partial^{\beta}(\nabla\cdot\textbf{v}_{a})]=0,$
(82)

where the continuity equation is used for the time derivatives of
concentration. Make the condition of the potentiality of the velocity field
$\textbf{v}_{a}=\nabla\phi_{a}$. Include it in equation (82) and find that
this equation is satisfied.

Hence, we obtain the simplified form of the pressure evolution equations,
where the traditional quantum Bohm potential is extracted:

$\partial_{t}p_{qf,B}^{\alpha\beta}+v_{B}^{\gamma}\partial_{\gamma}p_{qf,B}^{\alpha\beta}+p_{qf,B}^{\alpha\gamma}\partial_{\gamma}v_{B}^{\beta}+p_{qf,B}^{\beta\gamma}\partial_{\gamma}v_{B}^{\alpha}$
$+p_{qf,B}^{\alpha\beta}\partial_{\gamma}v_{B}^{\gamma}+\partial_{\gamma}Q^{\alpha\beta\gamma}_{qf,B}=0,$
(83)

and

$\partial_{t}p_{n}^{\alpha\beta}+v_{n}^{\gamma}\partial_{\gamma}p_{n}^{\alpha\beta}+p_{n}^{\alpha\gamma}\partial_{\gamma}v_{n}^{\beta}+p_{n}^{\beta\gamma}\partial_{\gamma}v_{n}^{\alpha}$
$+p_{n}^{\alpha\beta}\partial_{\gamma}v_{n}^{\gamma}+\partial_{\gamma}Q^{\alpha\beta\gamma}_{n}=0.$
(84)

Equation for the evolution of quantum-thermal part of the third rank tensor is
Andreev 2005 , Andreev 2007 :

$\partial_{t}Q_{qf}^{\alpha\beta\gamma}+\partial_{\delta}(v_{B}^{\delta}Q_{qf}^{\alpha\beta\gamma})+Q_{qf}^{\alpha\gamma\delta}\partial_{\delta}v_{B}^{\beta}+Q_{qf}^{\beta\gamma\delta}\partial_{\delta}v_{B}^{\alpha}+Q_{qf}^{\alpha\beta\delta}\partial_{\delta}v_{B}^{\gamma}$
$=\frac{\hbar^{2}}{4m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}\biggl{(}n_{B}\partial^{\delta}n_{B}+2n_{B}\partial^{\delta}n_{n})\biggr{)}$
$-\frac{1}{m}n_{B}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}V_{ext}+\tilde{F}_{B}^{\alpha\beta\gamma}+\tilde{F}_{B}^{\beta\gamma\alpha}+\tilde{F}_{B}^{\gamma\alpha\beta}$
$+\frac{1}{mn}(p_{qf,eff}^{\alpha\beta}\partial^{\delta}p_{qf,eff}^{\gamma\delta}+p_{qf,eff}^{\alpha\gamma}\partial^{\delta}p_{qf,eff}^{\beta\delta}+p_{qf,eff}^{\beta\gamma}\partial^{\delta}p_{qf,eff}^{\alpha\delta}),$
(85)

and

$\partial_{t}Q_{n}^{\alpha\beta\gamma}+\partial_{\delta}(v_{n}^{\delta}Q_{n}^{\alpha\beta\gamma})+Q_{n}^{\alpha\gamma\delta}\partial_{\delta}v_{n}^{\beta}+Q_{n}^{\beta\gamma\delta}\partial_{\delta}v_{n}^{\alpha}+Q_{n}^{\alpha\beta\delta}\partial_{\delta}v_{n}^{\gamma}$
$=\frac{\hbar^{2}}{2m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}\biggl{(}n_{n}\partial^{\delta}n_{n}+n_{n}\partial^{\delta}n_{B}\biggr{)}$
$-\frac{1}{m}n_{n}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}V_{ext}+\tilde{F}_{n}^{\alpha\beta\gamma}+\tilde{F}_{n}^{\beta\gamma\alpha}+\tilde{F}_{n}^{\gamma\alpha\beta}$
$+\frac{1}{mn}(p_{n,eff}^{\alpha\beta}\partial^{\delta}p_{n,eff}^{\gamma\delta}+p_{n,eff}^{\alpha\gamma}\partial^{\delta}p_{n,eff}^{\beta\delta}+p_{n,eff}^{\beta\gamma}\partial^{\delta}p_{n,eff}^{\alpha\delta}),$
(86)

where $\tilde{F}_{a}^{\alpha\beta\gamma}$ is not presented explicitly since
equations (66) and (67) show that required expressions are rather large.

Refs. Andreev PRA08 , Andreev LP 19 . Hydrodynamic model for fermions with
pressure evolution is derived in Refs. Andreev 2001 , Andreev 1912 , Andreev
LP 21 .

Terms proportional to
$p_{a,eff}^{\alpha\beta}\partial^{\delta}p_{a,eff}^{\gamma\delta}$ appears in
the pressure flux evolution equation, but it leads to the contribution beyond
the chosen approximation Tokatly PRB 99 , Tokatly PRB 00 .

Term containing the external potential
$-\frac{1}{m}n_{a}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}V_{ext}$
goes to zero for the parabolic trap. However, it can give some nontrivial
contribution for other form of potentials.

##  V BEC dynamics under the influence of the quantum fluctuations

Developed model shows that there is nontrivial evolution equation for the
pressure and the pressure flux of the BEC. Therefore, the well-known model of
BEC is extended in spite the fact that the kinetic pressure tensor is expected
to be equal to zero due to zero temperature. However, the quantum fluctuations
lead to the nonzero occupation numbers for the excited states.

If we need to consider pure BEC we need to drop the contribution of the normal
fluid in the model presented above. Therefore, let us summarize the BEC model
in parabolic traps:

$\partial_{t}n_{B}+\nabla\cdot(n_{B}\textbf{v}_{B})=0,$ (87)
$mn_{B}(\partial_{t}+\textbf{v}_{B}\cdot\nabla)v^{\alpha}_{B}+\partial_{\beta}(p_{qf}^{\alpha\beta}+T_{B}^{\alpha\beta})$
$+gn_{B}\partial^{\alpha}n_{B}+n_{B}\partial^{\alpha}V_{ext}=0,$ (88)
$\partial_{t}p_{qf}^{\alpha\beta}+v_{B}^{\gamma}\partial_{\gamma}p_{qf}^{\alpha\beta}+p_{qf}^{\alpha\gamma}\partial_{\gamma}v_{B}^{\beta}+p_{qf}^{\beta\gamma}\partial_{\gamma}v_{B}^{\alpha}$
$+p_{qf}^{\alpha\beta}\partial_{\gamma}v_{B}^{\gamma}+\partial_{\gamma}Q^{\alpha\beta\gamma}_{qf}=0,$
(89)

and

$\partial_{t}Q_{qf}^{\alpha\beta\gamma}+\partial_{\delta}(v^{\delta}Q_{qf}^{\alpha\beta\gamma})+Q_{qf}^{\alpha\gamma\delta}\partial_{\delta}v^{\beta}$
$+Q_{qf}^{\beta\gamma\delta}\partial_{\delta}v^{\alpha}+Q_{qf}^{\alpha\beta\delta}\partial_{\delta}v^{\gamma}=\frac{\hbar^{2}}{4m^{2}}g_{2}I_{0}^{\alpha\beta\gamma\delta}n\partial^{\delta}n.$
(90)

This simplified model is reported earlier in Refs. Andreev 2005 , Andreev 2007
, Andreev 2009 , where the dipole-dipole interaction is also considered. It
has been demonstrated that the quantum fluctuations gives mechanisms for the
instability of the small amplitude perturbations Andreev 2005 . Moreover, the
dipolar part of the quantum fluctuations creates conditions for the bright
soliton in the repulsive BECs Andreev 2009 . The developed in previous section
model provides proper generalization of earlier model giving the small
temperature contribution.

##  VI Conclusion

Revision of the two-fluid model for the finite temperature ultracold bosons
has been presented through the derivation of the number of particles, the
momentum, the momentum flux, and the third rank tensor balance equations. The
derivation has been based on the trace of the microscopic dynamics of quantum
particles via the application of the many-particle microscopic Schrodinger
equation. Hence, the microscopic Schrodinger equation determines the time
evolution for the macroscopic functions describing the collective motion of
bosons.

General equations have been derived for the arbitrary strength of interaction
and the arbitrary temperatures. The set of equations has been restricted by
the third rank kinematic tensor (the flux of pressure). The truncation is made
for the low temperatures weakly interacting bosons after the derivation of the
general structure of hydrodynamic equations. Therefore, the thermal part of
the fourth rank kinematic tensor has been taken equal to zero. Next, the terms
containing the interaction potential have been considered for the short-range
interaction. The small radius of interaction provides the small parameter for
the expansion. The expansion is made in the force field in the Euler equation,
the force tensor field in the momentum flux equation, and the third rank force
tensor in the pressure flux evolution equation. The first term in expansion on
the small interparticle distance has been considered in each expansion, which
corresponds to the first order by the interaction radius.

This model allows to gain the quantum fluctuations which is the essential
property of the BECs. Moreover, the interaction causing the quantum
fluctuations has been consider at the finite temperature.

The functions obtained in the first order by the interaction radius have been
expressed via the trace of two-particle functions. The two-particle functions
have been calculated for the weakly interacting bosons.

The single species of 0-spin bosons has been considered. Therefore, the single
fluid hydrodynamics has been derived. Next, it has been included that the
concentration of particles, the current of particles (the momentum density),
the momentum flux, and the current of the momentum flux are additive
functions. Consequently, they can be easily splitted on two parts: the BEC and
the normal fluid of bosons (the non BEC part). Hence, the two fluid model of
single species of bosons is obtained. This separation on two fluids has been
made in general form of equations.

##  VII Acknowledgements

Work is supported by the Russian Foundation for Basic Research (grant no.
20-02-00476). This paper has been supported by the RUDN University Strategic
Academic Leadership Program.

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