# Nonstationary Superconductivity: Quantum Dissipation and Time-Dependent Ginzburg-Landau Equation

Authors: Anatoly A. Barybin
Journal: Advances in Condensed Matter Physics
 (2011)

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## Abstract

Transport equations of the macroscopic superfluid dynamics are revised on the basis of a combination of the conventional (stationary) Ginzburg-Landau equation and Schrödinger's equation for the macroscopic wave function (often called the order parameter) by using the well-known Madelung-Feynman approach to representation of the quantum-mechanical equations in hydrodynamic form. Such an approach has given (a) three different contributions to the resulting chemical potentialμs
for the superfluid component, (b) a general hydrodynamic equation of superfluid motion, (c) the continuity equation for superfluid flow with a relaxation term involving the phenomenological parameters τGL and ξGL, (d) a new version of the time-dependent Ginzburg-Landau equation for the modulus of the order parameter which takes into account dissipation effects and reflects the charge conservation property for the superfluid component. The conventional Ginzburg-Landau equation also follows from our continuity equation as a particular case of stationarity. All the results obtained are mutually consistent within the scope of the chosen phenomenological description and, being model-neutral, applicable to both the low-Tc and high-Tc superconductors.


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## Body

## 1. Introduction

In superconductivity, the present efforts of scientists are mainly concentrated on studying the high-Tc superconductors. Yet, there are some theoretical problems in the low-Tc superconductivity left unsolved up to now. One of them (relating to thermoelectricity and having remained mysterious for many years because of great discrepancy between theory and experiments) has recently been explained by using the Fermi liquid description and charge imbalance notion [1].Another problem left unsolved from the past is associated with the nonequilibrium behavior of superfluid flow when time-dependent situations result in dissipative effects, which are non-existent in the stationary regime. In particular, many efforts were spent previously to convert the usual (stationary) Ginzburg-Landau equation into a time-dependent (nonstationary) form but without effect, unfortunately, because all such attempts resulted in an equation of charge non-conserving form.Before pursuing this point in more detail, it is useful to consider the similar problem relating to elaboration of the dissipative quantum theory for usual (non-superconductive) systems with damping.On the classical level, a variational principle was first applied to dissipative systems by Bateman [2] on the basis of doubling the system degrees of freedom. Bateman's classical approach was later imported to quantum field theory [3] and this method has been since then applied to a number of problems. However, for our purposes it is more interesting to consider another approach set forth below.Over sixty years ago, Caldirola [4] and Kanai [5] derived a quantum-mechanical equation for the wave function ψ. They have described the damped motion of a charged particle with charge e and mass m by introducing a phenomenological dissipation parameter λ. Another form of the nonlinear wave equation, known as the Schrödinger–Langevin equation, was obtained by Kostin [6] when examining the quantum motion of a Brownian particle.Afterwards, Wagner [7] carried out a comparative study of the Caldirola-Kanai and Schrödinger-Langevin equations on the basis of the hydrodynamic treatment, which was first suggested by Madelung [8]. In accordance with Madelung's picture, the wave function is represented as ψ=ρexp(iS/ℏ). In this case, the separation of Schrödinger's equation into the real and imaginary parts yields both the continuity equation,   (1)∂ρ∂t+∇⋅(ρv)=0,
and the hydrodynamic equation of motion,(2)∂v∂t+(v⋅∇)v=-1m∇(V+Q)-λv,
where (3)v=1m∇S,Q=-ℏ22m∇2ρ1/2ρ1/2.The quantum hydrodynamic equation (2) takes into account dissipative effects in the ordinary way by introducing a phenomenological relaxation term -λv. Besides the electric potential V, it contains the so-calledquantum potentialQ (both V and Q are measured in eV). In the classical limit ℏ→0, when the quantity S makes sense of the classical action, the quantum potential Q vanishes.Expressions (3) are valid only for the Schrödinger-Langevin-Kostin formulation [6], while the Caldirola-Kanai equations [4, 5] produce some extra loss factors for v and Q, exp(-λt) and exp(-2λt), respectively.As is shown by Wagner [7], in the dynamic regime with taking into account both the scalar potential φ=V/e and the vector potential A, when (in the SI system)(4)E=-∂A∂t-∇φ,B=∇×A,
expression (3) for the quantum potential Q remains unchanged. However, expression (3) for the hydrodynamic velocity v assumes the following form: (5)v=(∇S-eA)m.
Moreover, with allowance for v, the hydrodynamic equation (2) acquires some additional terms which, however, have no reasonable physical explanation [7].It should be noted that the quantum potential concept appearing in Madelung's formulation of quantum mechanics [8] was applied by a number of authors to solve different physical problems (e.g., see [9–12]) including quantum dissipation effects in Schrödinger-Langevin-Kostin formulation [13, 14].As applied to the superfluid dynamics in superconductors, the problem of quantum dissipation is treated in the form of the time-dependent Ginzburg-Landau (TDGL) equation (e.g., see [15]). Generalization of the usual Ginzburg-Landau (GL) stationary theory [16–18] to time-varying situations was the subject of much investigation (e.g., see reviews [19–21]). However, all known versions of the TDGL equation suffer from a general drawback: they have no charge conservation property, that is, do not provide the continuity equation for the superfluid flow (butnot for the total flow, superfluid and normal fluid together) (see the text after formula (38)).As will be shown below, unlike the nonsuperconductive systems with losses subject to (1) and (2), insertion of the quantum dissipation effect in the TDGL equation for superconductors gives rise to the continuity equation for superfluid flow in the form with nonzero right-hand part, whereas the equation of supercarrier motion involves no relaxation term like -λv. This inference results from applying Madelung's treatment to our version of the TDGL equation which results from combining Schrödinger's equation with the conventional Ginzburg-Landau equation.Section2 deals with such a phenomenological treatment based on the quantum-mechanical Madelung-Feynman approach to obtain the transport equations for superfluid motion. Our modified version of the TDGL equation provides a basis for deriving the desired forms of the continuity equation, the dynamic equation of motion, and the general expression for the total chemical potential of the superfluid component in dynamic regime.Section3 converts the continuity equation for superfluid flow into two forms corresponding to the stationary and time-dependent Ginzburg-Landau equations written, for the modulus of the order parameter.


## 2. Transport Equations of Macroscopic Superfluid Dynamics

### 2.1. Madelung-Feynman's Approach to the Quantum-Mechanical Equations in Hydrodynamic Form

Madelung [8] was the first to suggest the hydrodynamic form of quantum theory by decomposing Schrödinger's equation into the real and imaginary parts. As is noted in Section 1, Madelung's approach was first applied [4–7] to the dissipative quantum-mechanical systems described by the microscopic wave functionψ=ρexp(iS/ℏ), which gives rise to the customary hydrodynamic equations (1) and (2).Afterwards, Feynman [22, 23] extended this approach to superconductivity assuming the macroscopic wave function (often called theorder parameter) written in the following form (6)Ψ(r,t)=a(r,t)exp⁡⁡[iΘ(r,t)],
satisfies the ordinary Schrödinger equation to describe the superconducting condensate. The reasoning for such an assumption is that the macroscopic wave function describes a large number of zero-spin particles (Cooper's pairs), all doing precisely the same thing (strongly correlated motion). So the time rate of change in this function must be exactly the same as the rate for any single particle. This wave function, being complex-valued, is normalized so that |Ψ|2≡a2=Ns/2, where Ns is the superconducting electron density. The phase  Θ≡S/ℏ  is associated with the canonical (generalized) momentum of Cooper's pairs (7)P=2(mvs+eA),
by the known relation  P=ℏ∇Θ≡∇S [17, 18].Relation (7) is a generalization of expression (5) to Cooper's pairs with charge qs=2e and mass ms=2m, having velocity vs and concentration Ns/2.On these grounds, following Feynman [22, 23] we write Schrödinger's equation for the macroscopic wave function (6) in the conventional form (using the SI system) (8)iℏ∂Ψ∂t=[12ms(-iℏ∇-qsA)2+U]Ψ.
Here the potential energy, written as(9)U=qsφ+2μs0=2(eφ+μs0),
is determined not only by the electric potential φ but also by the chemical potential 2μs0 for Cooper's pairs in the condensate. A physical meaning of the chemical potential μs0 will be discussed later (see formula (49)). Its insertion into (9) is an essential feature of our treatment. This point distinguishes between our form (8) of Schrödinger's equation for themacroscopic wave function Ψ and the similar one for themicroscopic wave function ψ (with no insertion of the chemical potential into the potential energy (9), as it is usually done in the conventional quantum-mechanical theory).Substituting the expression (6) for Ψ in Schrödinger's equation (8) and separating the result into the real and imaginary parts, following Madelung [8], yield: (i)
from the imaginary part(10)∂Ns∂t+∇⋅(Nsvs)=0,(ii)
from the real part
(11)μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t=0.Here, in addition toμs0, there is an extra contribution to the total chemical potential
(12)μs1=-ℏ28m∇2Ns1/2Ns1/2.As is seen from expression (12), the additional chemical potential μs1 is due to a spatial nonuniformity of the superfluid density Ns, which takes place, for example, in the vicinity of any vortex line or the NS-interface.Relation (11) includes also the term mvs2/2 caused by superfluid flow with velocity vs. Let us apply the gradient operation to (11) by using the known identity [24] (13)∇mvs22=mvs⋅∇vs+mvs×(∇×vs)
and the total derivative of vs with respect to time(14)dvsdt≡∂vs∂t+vs⋅∇vs=∂vs∂t+∇vs22-vs×(∇×vs).Employing the following expression forvs resulted from (7):(15)vs=P2m-(em)A≡(ℏ2m)∇Θ-(em)A
and expression (4) for B, we obtain(16)∇×vs=-em∇×A=-emB.After the above transformations applied to (11), we arrive at the following equation of superfluid motion:  (17)mdvsdt=e(E+vs×B)-∇(μs0+μs1).A comparison of (2) and (4) with (17) shows that the chemical potential (12), being doubled, can be interpreted, by analogy with formula (3) for Q, as the quantum potential of Cooper's pairs (ρs=qsNs/2=eNs) (18)Qs≡2μs1=-ℏ22ms∇2ρs1/2ρs1/2.The continuity equation (10) obtained at this stage of our examination does not take into account the dynamic processes of breaking and creating for Cooper's pairs. This point should not be surprising because such processes are necessarily beyond the ordinary Schrödinger equation (8). This equation has been used above as an initial one to obtain (10) and (11) and it describes only the dynamic processes ofnondissipative character. In the phenomenological theory of nonstationary superconductivity, the dissipation effects are usually attained by using the time-dependent Ginzburg-Landau (TDGL) equation [18–21]. The TDGL equation is a generalization of the ordinary GL equation (originally obtained [16] for the steady-state nonuniform superconductors) to time-varying situations.


### 2.2. Combined Approach to the Schrödinger and Ginzburg-Landau Equations

In their pioneer work [16], Ginzburg and Landau began searching phenomenologically a proper stationary form for the free energy functional,(19)Fs=∫fs(Ψ,Ψ*,A)dV,
where its density fs depends on the vector potential A, as well as on the complex order parameter Ψ given in the form of (6). To find Ψ(r), (19) was derived by requiring ℱs to be minimum with respect to variation of Ψ*, that is, δℱs/δΨ*=0. Such an operation has led to the usual GL equation [16, 18](20)∂fs∂Ψ*≡[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ=0,
where the zero equality is required to ensure stationary conditions. The parameters α and β introduced phenomenologically are often associated with two other measurable temperature-dependent quantities: theequilibrium superfluid density Ns°(T)=2|Ψ°|2 and the thermodynamic critical magnetic field Hc(T) by means of the following relations (written in the SI system of units) [18, 25]:(21)α(T)=-2μ0Hc2(T)Ns°(T),β(T)=2|α(T)|Ns°(T).
Here (and also below) the superscript ° corresponds to the equilibrium state and μ0 is the vacuum permeability.In the literature, there are different versions of the time-dependent generalization of the stationary GL equation (20). A common feature of all the known phenomenological approaches is the assumption that the function Ψ(r,t) tends to its stationary value (corresponding to minimum of the free energy functional ℱs) at a rate governed by the slope of the density fs(Ψ,Ψ*,A) in the functional space, that is, by the value -∂fs/∂Ψ* which is now other than zero, unlike (20). This way has led to the following general form of the TDGL equation (e.g., see [15, 19, 20, 26]):(22)(∂∂t-2μiℏ)Ψ=-1ℏγ∂fs∂Ψ*≡-1ℏγ[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ.The right-hand part of (22) includes the relaxation parameter γ introduced phenomenologically and the functional derivative ∂fs/∂Ψ* in the form of (20) but not being equal to zero under nonequilibrium conditions. Second term on the left reduces to the following time dependence Ψ°(t)=|Ψ°|exp(-2iμt/ℏ) in equilibrium when γ→∞.Microscopic theory of the time-dependent superconductivity meets with serious complications arising from existence of the energy gap and singularity in the density of states at the gap edges. For this reason, the TDGL equation was derived by Gor’kov and Eliashberg [27] only for the gapless regime with predominant effect of paramagnetic impurities. Schmid [28] has obtained the analogous equation valid only near the transition temperature Tc and proposed that μ appearing in (22) should be considered as the electrochemical potential. A detailed discussion of the time-dependent problems can be found in a review [20].All versions of the TDGL equation have the left-hand side similar to that of (22). As is shown below, such a structure of the TDGL equation is not of charge-conservation character because it cannot provide the continuity equation for the superfluid motion (butnot for the total, normal and superfluid, motion which always conserves the charge).Our idea of obtaining another version of the time-dependent GL equation is based on an analogy with the conventional way in deriving the kinetic equation for the distribution function to describe the nonstationary behavior of carriers in semiconductors and normal metals.As is well-known (e.g., see books [29, 30]), in kinetics theory the time rate of change in the distribution function f(r,t) is taken to be divided into two parts: (i) the field contribution(∂f/∂t)field caused by the coherent action of electromagnetic fields and (ii) thedissipation contribution(∂f/∂t)diss caused by the incoherent relaxation processes of electron scattering and known as the collision integral, so that(23)∂f∂t=(∂f∂t)field+(∂f∂t)diss,
discarding the actual expressions for the two terms in (23), which can be found in the literature (see [29, 30]).In accordance with the line of our reasoning, the similar approach can be phenomenologically applied to the macroscopic wave functionΨ(r,t) of the superconducting state, so that its time rate of change is represented, by analogy with expression (23), as a sum of the similar two contributions:(24)∂Ψ∂t=(∂Ψ∂t)field+(∂Ψ∂t)diss.Thefield contribution is assumed to obey Schrödinger's equation (8) rewritten in the following form:(25)(∂Ψ∂t)field=1iℏĤΨ,
where, in accordance with (8) and (9), the Hamiltonian operator for Cooper's pairs is(26)Ĥ=12ms(-iℏ∇-qsA)2+qsφ+2μs0.Thedissipation contribution is taken to satisfy the following equation:(27)(∂Ψ∂t)diss=-1ℏγ∂fs∂Ψ*,
whose right-hand part coincides with that of (22), where the functional derivative(28)∂fs∂Ψ*=[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ
has the form of (20) theoretically substantiated by Ginzburg and Landau [16] but different from zero under nonequilibrium conditions.Substitution of relations (25)–(28) into (24) gives a desired novel version of the nonstationary equation, giving rise to both the usual GL equation (see (62)–(64)) and the TDGL equation (see (67)–(69)):(29)∂Ψ∂t-1iℏ[12ms(-iℏ∇-qsA)2+qsφ+2μs0]Ψ=-1ℏγ[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ.This equation, being time-dependent and space-nonuniform, allows for dissipation processes characterized by the phenomenological parameterγ, as well as by α and β. In comparison with the conventional equation (22) existing now in the literature [15, 19, 20, 26], (29) contains in the left-hand side the additional terms involving the space operator ∇ and the electromagnetic potentials A and φ, provided the quantity μ in (22) means μs0. It is these terms that turn out to be of crucial importance in obtaining below the unambiguous and consistent results, including: (i) the continuity equation (36) with nonzero relaxation term instead of (10) and (ii) the superfluid motion equation (52) instead of (17).Phenomenological parametersα,β, and γ are often replaced by others [15, 25]:(30)τGL(T)=ℏγ(T)|α(T)|≡ℏγ(T)2Ns°(T)μ0Hc2(T),ξGL2(T)=ℏ22ms|α(T)|≡ℏ28mNs°(T)μ0Hc2(T),ηGL(T)=β(T)|α(T)|≡2Ns°(T)
where the last equalities are written by applying formula (21). The quantities τGL(T) and ξGL(T) are known as the temperature-dependent relaxation time and coherence length of the GL theory [17–21, 27].In the case of employing parameters (30), our initial equation (29) takes the following form:(31)iℏ∂Ψ∂t-ĤΨ=iℏτGLR̂Ψ.Here the Hamiltonian operatorĤ (nondissipative) is given by (26) and the relaxation operator R̂ (dissipative) on the right is now defined as(32)R̂=ξGL2(∇+2eiℏA)2-ηGL|Ψ|2+1.It is easy to see that (31) is converted either into the ordinary GL equation (20) when τGL→0 (relaxation processes are dominant) or into Schrödinger's equation (8) when τGL-1→0 (relaxation processes are negligible). Something close to our result was remarked by Abrahams and Tsuneto [26].It is very important to note that the right-hand side of (31) can exactly be reduced to the form obtained on the basis of microscopic theory for gapless superconductors by Gor'kov and Eliashberg [27] (see also works [18–20]) only if the quantity DGL≡ξGL2/τGL=ℏ/4mγ is identified with the normal-state diffusion coefficient D=vF2τ/3.The left-hand and right-hand sides of (31) are, respectively, transformed by using the macroscopic wave function in the form of Ψ=Ns/2exp(iΘ) (see (6)) into the following: (33)iℏ∂Ψ∂t-ĤΨ=-[2(μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t)-iℏ2Ns(∂Ns∂t+∇⋅(Nsvs))(μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t)]Ψ,(34)iℏτGLR̂Ψ=-iℏτGL(G+ξGL22miℏ∇⋅(Nsvs)Ns)Ψ.
Here we have used formula (7) the relations (vs·∇a)/a=(∇·Nsvs)/2Ns and ∇·vs=0 (see (44)), and also introduced a quantity(35)G=ξGL28mℏ2(mvs22+μs1)+NsNs°-1.After substituting (33) and (34) into (31) and separating out the imaginary and real parts, we obtain: (i)
the continuity equation for superfluid motion(36)∂Ns∂t+∇⋅(Nsvs)=-2GτGLNs,(ii)
the relation for chemical potentials(37)μs0+μs1+μs2+eφ+mvs22+ℏ2∂Θ∂t=0.Here, in addition to the chemical potentialsμs0 and μs1 we have introduced another contribution(38)μs2=-ℏ4γNs∇⋅(Nsvs)≡-ξGL2τGLme∇⋅JsNs.The chemical potentialμs2, just like μs1, is due to the nonuniform spatial distribution of supercarriers but, unlike μs1, is solely of dissipative nature. It is interesting to notice that μs2 fully coincides with the quantity 2(μp-μ) introduced by Rieger et al. [31] to describe mutual conversion of the superfluid and normal fluid components in the vicinity of NS-interfaces.It is pertinent to stress that if we employed the conventional TDGL equation (22) instead of our equations (31)-(32) as a basis for the transformations done above, it would lead us to the different result: there would be neither the term ∇·(Nsvs) in (36) nor the terms μs1, eφ and mvs2/2 in (37). It is the former fact that allows us to draw the conclusion about the charge-nonconservation character of all the existing versions of the TDGL equation, which have the left-hand side similar to that of (22). The above-mentioned arguments count in favor of our version of the TDGL equation given by (29) or (31)-(32).If we define the total chemical potential of the superfluid component in a dynamic regime as the sum of the four contributions (see (12) and (38)):(39)μΣ=μs0+μs1+μs2+mvs22=μs+mvs22≡(μs0-ℏ28m∇2Ns1/2Ns1/2-ξGL2τGLme∇⋅JsNs)+mvs22,
then formula (37) may be rewritten in two ways(40)Φ≡eφ+ℏ2∂Θ∂t=-(μs+mvs22)≡-μΣ,(41)ζΣ≡ζs+mvs22≡μΣ+eφ=-ℏ2∂Θ∂t,
where ζs=μs+eφ is the electrochemical potential of the superfluid component.Equation (40) shows that the total chemical potential μΣ always varies oppositely in phase with respect to the so-calledgauge-invariant potentialΦ, being equal to it in magnitude. Besides Φ, the superconductivity theory deals with another gauge-invariant quantity, namely, the superfluid velocity vs [17, 18]. The use of them allows us, following Stephen [32], to introduce the gauge-invariant electromagnetic potentials (marked with subscript g)(42)Ag=-mvse,φg=Φe.In this case, the electric field (4) can be rewritten in the gauge-invariant form: (43)E=-∂Ag∂t-∇φg.As is known from the classical electrodynamics [33], in order for the gradient term in (43) to be actually the potential field Ep=-∇φg with the scalar potential φg obeying Poisson's equation, it is necessary to employ the Coulomb gauge (∇·Ag=0). Hence, using formulae (16) and (42), we arrive at the following requirements imposed on the superfluid velocity:(44)∇⋅vs=0,∇×vs=-(em)B≠0.Formula (44) expresses the property ofsolenoidality inherent in the superfluid flow. The universal condition ∇·vs=0 ensures that the component of superfluid velocity normal to any boundary vanishes (n·vs=0). In the vicinity of NS-interface there is a conversion of the normal current with n·vn≠0 into the supercurrent with n·vs=0 and nonuniform distribution of Ns.According to (41), the electrochemical potential ζs=μs+eφ is determined by the rate of change in the phase Θ of the order parameter Ψ. One should recall that all the energy values are measured with respect to the unperturbed bottom of the energy band (ℰb°=0). Then, by definition, in equilibrium μs°≡ζs°-ℰb°=ζs°. There being the potential electric field Ep=-∇φ with the London gauge ∇·A=0, the band bottom suffers an offset from ℰb°=0, so that ℰb=eφ and μs≡ζs-ℰb=ζs-eφ.Let us clarify a sense of the initial chemical potentialμs0 appearing in (9) and find its relationship to the Fermi energy ℰF. First consider the equilibrium superconducting state with no superfluid flow (vs=0) when the chemical potentials μseq and μneq are both equal to ℰF. Then the Gibbs free energy per unit volume is (45)G(vs=0)=μseqNseq+μneqNneq=EFN.
In the presence of the stationary superfluid flow (vs≠0), the equilibrium is not disturbed, yet both the chemical potentials and electron densities become altered so that μs°=μn°≠ℰF and Ns°+Nn°=Nseq+Nneq=N. Then(46)G(vs≠0)=μs°Ns°+μn°Nn°=μs°N.
It is evident that the alteration in 𝒢 is brought about by an increase in the kinetic energy of supercarriers(47)G(vs≠0)-G(vs=0)=mvs22Ns∘.
From the above reasoning it follows that in the case of stationary superfluid flow,(48)μs∘=EF+mvs22Ns°N=μn°.
It is this quantity taking account of the superfluid motion that can be identified with the above introduced value μs0+mvs2/2, then (49)μs0=μs°-mvs22=EF-mvs22Nn°N.Equations (48) and (49) are similar to those given by Ginzburg and Zharkov [34, formula (9.8) and (9.9)], the addition to ℰF in (48) being associated with the Bernoulli potential, which was observed experimentally [35, 36].Thus, we can  rewrite the total chemical potential (39) in the following form: (50)μΣ=μs°+δμs,
where the stationary value μs∘ and its perturbation δμs are defined as follows (see (12), (38), and (49)): (51)μs°=μs0+mvs22,δμs=μs1+μs2≡-ℏ28m∇2Ns1/2Ns1/2-ξGL2τGLme∇⋅JsNs.The dynamic additionδμs to the equilibrium chemical potential μs°=μn° reflects an appearance of the normal current Jn=eNnvn together with the nonuniform superfluid density δNs=Ns-Ns° as a result of the time-dependent and space-dependent external actions.Applying the gradient operation to (37) and carrying out the same procedure, as applied to formula (11), we finally obtain instead of (17) the required hydrodynamic equation of superfluid motion:(52)mdvsdt=e(E+vs×B)-∇μs.
Here, the resulting chemical potential μs of the superfluid component consists of three contributions,(53)μs=μs0+μs1+μs2,
defined by (49), (12), and (38), respectively.In comparison with (17), (52) contains, besides μs0 and μs1, the chemical potential μs2, (38). The latter reflects influence of peculiar dissipative effects specified by the parameter γ entering originally into the right-hand side of the time-dependent equation (29). This influence differs from that expressed by the dissipative term -λv in (2) and, being proportional to the gradient ∇Ns (since ∇·(Nsvs)=vs·∇Ns owing to ∇·vs=0), it is of specific diffusion character.


## 2.1. Madelung-Feynman's Approach to the Quantum-Mechanical Equations in Hydrodynamic Form

Madelung [8] was the first to suggest the hydrodynamic form of quantum theory by decomposing Schrödinger's equation into the real and imaginary parts. As is noted in Section 1, Madelung's approach was first applied [4–7] to the dissipative quantum-mechanical systems described by the microscopic wave functionψ=ρexp(iS/ℏ), which gives rise to the customary hydrodynamic equations (1) and (2).Afterwards, Feynman [22, 23] extended this approach to superconductivity assuming the macroscopic wave function (often called theorder parameter) written in the following form (6)Ψ(r,t)=a(r,t)exp⁡⁡[iΘ(r,t)],
satisfies the ordinary Schrödinger equation to describe the superconducting condensate. The reasoning for such an assumption is that the macroscopic wave function describes a large number of zero-spin particles (Cooper's pairs), all doing precisely the same thing (strongly correlated motion). So the time rate of change in this function must be exactly the same as the rate for any single particle. This wave function, being complex-valued, is normalized so that |Ψ|2≡a2=Ns/2, where Ns is the superconducting electron density. The phase  Θ≡S/ℏ  is associated with the canonical (generalized) momentum of Cooper's pairs (7)P=2(mvs+eA),
by the known relation  P=ℏ∇Θ≡∇S [17, 18].Relation (7) is a generalization of expression (5) to Cooper's pairs with charge qs=2e and mass ms=2m, having velocity vs and concentration Ns/2.On these grounds, following Feynman [22, 23] we write Schrödinger's equation for the macroscopic wave function (6) in the conventional form (using the SI system) (8)iℏ∂Ψ∂t=[12ms(-iℏ∇-qsA)2+U]Ψ.
Here the potential energy, written as(9)U=qsφ+2μs0=2(eφ+μs0),
is determined not only by the electric potential φ but also by the chemical potential 2μs0 for Cooper's pairs in the condensate. A physical meaning of the chemical potential μs0 will be discussed later (see formula (49)). Its insertion into (9) is an essential feature of our treatment. This point distinguishes between our form (8) of Schrödinger's equation for themacroscopic wave function Ψ and the similar one for themicroscopic wave function ψ (with no insertion of the chemical potential into the potential energy (9), as it is usually done in the conventional quantum-mechanical theory).Substituting the expression (6) for Ψ in Schrödinger's equation (8) and separating the result into the real and imaginary parts, following Madelung [8], yield: (i)
from the imaginary part(10)∂Ns∂t+∇⋅(Nsvs)=0,(ii)
from the real part
(11)μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t=0.Here, in addition toμs0, there is an extra contribution to the total chemical potential
(12)μs1=-ℏ28m∇2Ns1/2Ns1/2.As is seen from expression (12), the additional chemical potential μs1 is due to a spatial nonuniformity of the superfluid density Ns, which takes place, for example, in the vicinity of any vortex line or the NS-interface.Relation (11) includes also the term mvs2/2 caused by superfluid flow with velocity vs. Let us apply the gradient operation to (11) by using the known identity [24] (13)∇mvs22=mvs⋅∇vs+mvs×(∇×vs)
and the total derivative of vs with respect to time(14)dvsdt≡∂vs∂t+vs⋅∇vs=∂vs∂t+∇vs22-vs×(∇×vs).Employing the following expression forvs resulted from (7):(15)vs=P2m-(em)A≡(ℏ2m)∇Θ-(em)A
and expression (4) for B, we obtain(16)∇×vs=-em∇×A=-emB.After the above transformations applied to (11), we arrive at the following equation of superfluid motion:  (17)mdvsdt=e(E+vs×B)-∇(μs0+μs1).A comparison of (2) and (4) with (17) shows that the chemical potential (12), being doubled, can be interpreted, by analogy with formula (3) for Q, as the quantum potential of Cooper's pairs (ρs=qsNs/2=eNs) (18)Qs≡2μs1=-ℏ22ms∇2ρs1/2ρs1/2.The continuity equation (10) obtained at this stage of our examination does not take into account the dynamic processes of breaking and creating for Cooper's pairs. This point should not be surprising because such processes are necessarily beyond the ordinary Schrödinger equation (8). This equation has been used above as an initial one to obtain (10) and (11) and it describes only the dynamic processes ofnondissipative character. In the phenomenological theory of nonstationary superconductivity, the dissipation effects are usually attained by using the time-dependent Ginzburg-Landau (TDGL) equation [18–21]. The TDGL equation is a generalization of the ordinary GL equation (originally obtained [16] for the steady-state nonuniform superconductors) to time-varying situations.


## 2.2. Combined Approach to the Schrödinger and Ginzburg-Landau Equations

In their pioneer work [16], Ginzburg and Landau began searching phenomenologically a proper stationary form for the free energy functional,(19)Fs=∫fs(Ψ,Ψ*,A)dV,
where its density fs depends on the vector potential A, as well as on the complex order parameter Ψ given in the form of (6). To find Ψ(r), (19) was derived by requiring ℱs to be minimum with respect to variation of Ψ*, that is, δℱs/δΨ*=0. Such an operation has led to the usual GL equation [16, 18](20)∂fs∂Ψ*≡[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ=0,
where the zero equality is required to ensure stationary conditions. The parameters α and β introduced phenomenologically are often associated with two other measurable temperature-dependent quantities: theequilibrium superfluid density Ns°(T)=2|Ψ°|2 and the thermodynamic critical magnetic field Hc(T) by means of the following relations (written in the SI system of units) [18, 25]:(21)α(T)=-2μ0Hc2(T)Ns°(T),β(T)=2|α(T)|Ns°(T).
Here (and also below) the superscript ° corresponds to the equilibrium state and μ0 is the vacuum permeability.In the literature, there are different versions of the time-dependent generalization of the stationary GL equation (20). A common feature of all the known phenomenological approaches is the assumption that the function Ψ(r,t) tends to its stationary value (corresponding to minimum of the free energy functional ℱs) at a rate governed by the slope of the density fs(Ψ,Ψ*,A) in the functional space, that is, by the value -∂fs/∂Ψ* which is now other than zero, unlike (20). This way has led to the following general form of the TDGL equation (e.g., see [15, 19, 20, 26]):(22)(∂∂t-2μiℏ)Ψ=-1ℏγ∂fs∂Ψ*≡-1ℏγ[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ.The right-hand part of (22) includes the relaxation parameter γ introduced phenomenologically and the functional derivative ∂fs/∂Ψ* in the form of (20) but not being equal to zero under nonequilibrium conditions. Second term on the left reduces to the following time dependence Ψ°(t)=|Ψ°|exp(-2iμt/ℏ) in equilibrium when γ→∞.Microscopic theory of the time-dependent superconductivity meets with serious complications arising from existence of the energy gap and singularity in the density of states at the gap edges. For this reason, the TDGL equation was derived by Gor’kov and Eliashberg [27] only for the gapless regime with predominant effect of paramagnetic impurities. Schmid [28] has obtained the analogous equation valid only near the transition temperature Tc and proposed that μ appearing in (22) should be considered as the electrochemical potential. A detailed discussion of the time-dependent problems can be found in a review [20].All versions of the TDGL equation have the left-hand side similar to that of (22). As is shown below, such a structure of the TDGL equation is not of charge-conservation character because it cannot provide the continuity equation for the superfluid motion (butnot for the total, normal and superfluid, motion which always conserves the charge).Our idea of obtaining another version of the time-dependent GL equation is based on an analogy with the conventional way in deriving the kinetic equation for the distribution function to describe the nonstationary behavior of carriers in semiconductors and normal metals.As is well-known (e.g., see books [29, 30]), in kinetics theory the time rate of change in the distribution function f(r,t) is taken to be divided into two parts: (i) the field contribution(∂f/∂t)field caused by the coherent action of electromagnetic fields and (ii) thedissipation contribution(∂f/∂t)diss caused by the incoherent relaxation processes of electron scattering and known as the collision integral, so that(23)∂f∂t=(∂f∂t)field+(∂f∂t)diss,
discarding the actual expressions for the two terms in (23), which can be found in the literature (see [29, 30]).In accordance with the line of our reasoning, the similar approach can be phenomenologically applied to the macroscopic wave functionΨ(r,t) of the superconducting state, so that its time rate of change is represented, by analogy with expression (23), as a sum of the similar two contributions:(24)∂Ψ∂t=(∂Ψ∂t)field+(∂Ψ∂t)diss.Thefield contribution is assumed to obey Schrödinger's equation (8) rewritten in the following form:(25)(∂Ψ∂t)field=1iℏĤΨ,
where, in accordance with (8) and (9), the Hamiltonian operator for Cooper's pairs is(26)Ĥ=12ms(-iℏ∇-qsA)2+qsφ+2μs0.Thedissipation contribution is taken to satisfy the following equation:(27)(∂Ψ∂t)diss=-1ℏγ∂fs∂Ψ*,
whose right-hand part coincides with that of (22), where the functional derivative(28)∂fs∂Ψ*=[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ
has the form of (20) theoretically substantiated by Ginzburg and Landau [16] but different from zero under nonequilibrium conditions.Substitution of relations (25)–(28) into (24) gives a desired novel version of the nonstationary equation, giving rise to both the usual GL equation (see (62)–(64)) and the TDGL equation (see (67)–(69)):(29)∂Ψ∂t-1iℏ[12ms(-iℏ∇-qsA)2+qsφ+2μs0]Ψ=-1ℏγ[12ms(-iℏ∇-qsA)2+α+β|Ψ|2]Ψ.This equation, being time-dependent and space-nonuniform, allows for dissipation processes characterized by the phenomenological parameterγ, as well as by α and β. In comparison with the conventional equation (22) existing now in the literature [15, 19, 20, 26], (29) contains in the left-hand side the additional terms involving the space operator ∇ and the electromagnetic potentials A and φ, provided the quantity μ in (22) means μs0. It is these terms that turn out to be of crucial importance in obtaining below the unambiguous and consistent results, including: (i) the continuity equation (36) with nonzero relaxation term instead of (10) and (ii) the superfluid motion equation (52) instead of (17).Phenomenological parametersα,β, and γ are often replaced by others [15, 25]:(30)τGL(T)=ℏγ(T)|α(T)|≡ℏγ(T)2Ns°(T)μ0Hc2(T),ξGL2(T)=ℏ22ms|α(T)|≡ℏ28mNs°(T)μ0Hc2(T),ηGL(T)=β(T)|α(T)|≡2Ns°(T)
where the last equalities are written by applying formula (21). The quantities τGL(T) and ξGL(T) are known as the temperature-dependent relaxation time and coherence length of the GL theory [17–21, 27].In the case of employing parameters (30), our initial equation (29) takes the following form:(31)iℏ∂Ψ∂t-ĤΨ=iℏτGLR̂Ψ.Here the Hamiltonian operatorĤ (nondissipative) is given by (26) and the relaxation operator R̂ (dissipative) on the right is now defined as(32)R̂=ξGL2(∇+2eiℏA)2-ηGL|Ψ|2+1.It is easy to see that (31) is converted either into the ordinary GL equation (20) when τGL→0 (relaxation processes are dominant) or into Schrödinger's equation (8) when τGL-1→0 (relaxation processes are negligible). Something close to our result was remarked by Abrahams and Tsuneto [26].It is very important to note that the right-hand side of (31) can exactly be reduced to the form obtained on the basis of microscopic theory for gapless superconductors by Gor'kov and Eliashberg [27] (see also works [18–20]) only if the quantity DGL≡ξGL2/τGL=ℏ/4mγ is identified with the normal-state diffusion coefficient D=vF2τ/3.The left-hand and right-hand sides of (31) are, respectively, transformed by using the macroscopic wave function in the form of Ψ=Ns/2exp(iΘ) (see (6)) into the following: (33)iℏ∂Ψ∂t-ĤΨ=-[2(μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t)-iℏ2Ns(∂Ns∂t+∇⋅(Nsvs))(μs0+μs1+eφ+mvs22+ℏ2∂Θ∂t)]Ψ,(34)iℏτGLR̂Ψ=-iℏτGL(G+ξGL22miℏ∇⋅(Nsvs)Ns)Ψ.
Here we have used formula (7) the relations (vs·∇a)/a=(∇·Nsvs)/2Ns and ∇·vs=0 (see (44)), and also introduced a quantity(35)G=ξGL28mℏ2(mvs22+μs1)+NsNs°-1.After substituting (33) and (34) into (31) and separating out the imaginary and real parts, we obtain: (i)
the continuity equation for superfluid motion(36)∂Ns∂t+∇⋅(Nsvs)=-2GτGLNs,(ii)
the relation for chemical potentials(37)μs0+μs1+μs2+eφ+mvs22+ℏ2∂Θ∂t=0.Here, in addition to the chemical potentialsμs0 and μs1 we have introduced another contribution(38)μs2=-ℏ4γNs∇⋅(Nsvs)≡-ξGL2τGLme∇⋅JsNs.The chemical potentialμs2, just like μs1, is due to the nonuniform spatial distribution of supercarriers but, unlike μs1, is solely of dissipative nature. It is interesting to notice that μs2 fully coincides with the quantity 2(μp-μ) introduced by Rieger et al. [31] to describe mutual conversion of the superfluid and normal fluid components in the vicinity of NS-interfaces.It is pertinent to stress that if we employed the conventional TDGL equation (22) instead of our equations (31)-(32) as a basis for the transformations done above, it would lead us to the different result: there would be neither the term ∇·(Nsvs) in (36) nor the terms μs1, eφ and mvs2/2 in (37). It is the former fact that allows us to draw the conclusion about the charge-nonconservation character of all the existing versions of the TDGL equation, which have the left-hand side similar to that of (22). The above-mentioned arguments count in favor of our version of the TDGL equation given by (29) or (31)-(32).If we define the total chemical potential of the superfluid component in a dynamic regime as the sum of the four contributions (see (12) and (38)):(39)μΣ=μs0+μs1+μs2+mvs22=μs+mvs22≡(μs0-ℏ28m∇2Ns1/2Ns1/2-ξGL2τGLme∇⋅JsNs)+mvs22,
then formula (37) may be rewritten in two ways(40)Φ≡eφ+ℏ2∂Θ∂t=-(μs+mvs22)≡-μΣ,(41)ζΣ≡ζs+mvs22≡μΣ+eφ=-ℏ2∂Θ∂t,
where ζs=μs+eφ is the electrochemical potential of the superfluid component.Equation (40) shows that the total chemical potential μΣ always varies oppositely in phase with respect to the so-calledgauge-invariant potentialΦ, being equal to it in magnitude. Besides Φ, the superconductivity theory deals with another gauge-invariant quantity, namely, the superfluid velocity vs [17, 18]. The use of them allows us, following Stephen [32], to introduce the gauge-invariant electromagnetic potentials (marked with subscript g)(42)Ag=-mvse,φg=Φe.In this case, the electric field (4) can be rewritten in the gauge-invariant form: (43)E=-∂Ag∂t-∇φg.As is known from the classical electrodynamics [33], in order for the gradient term in (43) to be actually the potential field Ep=-∇φg with the scalar potential φg obeying Poisson's equation, it is necessary to employ the Coulomb gauge (∇·Ag=0). Hence, using formulae (16) and (42), we arrive at the following requirements imposed on the superfluid velocity:(44)∇⋅vs=0,∇×vs=-(em)B≠0.Formula (44) expresses the property ofsolenoidality inherent in the superfluid flow. The universal condition ∇·vs=0 ensures that the component of superfluid velocity normal to any boundary vanishes (n·vs=0). In the vicinity of NS-interface there is a conversion of the normal current with n·vn≠0 into the supercurrent with n·vs=0 and nonuniform distribution of Ns.According to (41), the electrochemical potential ζs=μs+eφ is determined by the rate of change in the phase Θ of the order parameter Ψ. One should recall that all the energy values are measured with respect to the unperturbed bottom of the energy band (ℰb°=0). Then, by definition, in equilibrium μs°≡ζs°-ℰb°=ζs°. There being the potential electric field Ep=-∇φ with the London gauge ∇·A=0, the band bottom suffers an offset from ℰb°=0, so that ℰb=eφ and μs≡ζs-ℰb=ζs-eφ.Let us clarify a sense of the initial chemical potentialμs0 appearing in (9) and find its relationship to the Fermi energy ℰF. First consider the equilibrium superconducting state with no superfluid flow (vs=0) when the chemical potentials μseq and μneq are both equal to ℰF. Then the Gibbs free energy per unit volume is (45)G(vs=0)=μseqNseq+μneqNneq=EFN.
In the presence of the stationary superfluid flow (vs≠0), the equilibrium is not disturbed, yet both the chemical potentials and electron densities become altered so that μs°=μn°≠ℰF and Ns°+Nn°=Nseq+Nneq=N. Then(46)G(vs≠0)=μs°Ns°+μn°Nn°=μs°N.
It is evident that the alteration in 𝒢 is brought about by an increase in the kinetic energy of supercarriers(47)G(vs≠0)-G(vs=0)=mvs22Ns∘.
From the above reasoning it follows that in the case of stationary superfluid flow,(48)μs∘=EF+mvs22Ns°N=μn°.
It is this quantity taking account of the superfluid motion that can be identified with the above introduced value μs0+mvs2/2, then (49)μs0=μs°-mvs22=EF-mvs22Nn°N.Equations (48) and (49) are similar to those given by Ginzburg and Zharkov [34, formula (9.8) and (9.9)], the addition to ℰF in (48) being associated with the Bernoulli potential, which was observed experimentally [35, 36].Thus, we can  rewrite the total chemical potential (39) in the following form: (50)μΣ=μs°+δμs,
where the stationary value μs∘ and its perturbation δμs are defined as follows (see (12), (38), and (49)): (51)μs°=μs0+mvs22,δμs=μs1+μs2≡-ℏ28m∇2Ns1/2Ns1/2-ξGL2τGLme∇⋅JsNs.The dynamic additionδμs to the equilibrium chemical potential μs°=μn° reflects an appearance of the normal current Jn=eNnvn together with the nonuniform superfluid density δNs=Ns-Ns° as a result of the time-dependent and space-dependent external actions.Applying the gradient operation to (37) and carrying out the same procedure, as applied to formula (11), we finally obtain instead of (17) the required hydrodynamic equation of superfluid motion:(52)mdvsdt=e(E+vs×B)-∇μs.
Here, the resulting chemical potential μs of the superfluid component consists of three contributions,(53)μs=μs0+μs1+μs2,
defined by (49), (12), and (38), respectively.In comparison with (17), (52) contains, besides μs0 and μs1, the chemical potential μs2, (38). The latter reflects influence of peculiar dissipative effects specified by the parameter γ entering originally into the right-hand side of the time-dependent equation (29). This influence differs from that expressed by the dissipative term -λv in (2) and, being proportional to the gradient ∇Ns (since ∇·(Nsvs)=vs·∇Ns owing to ∇·vs=0), it is of specific diffusion character.


## 3. Continuity Equation and Time-Dependent Ginzburg-Landau Equation

Let us return to our result in the form of (36) giving the continuity equation for the superfluid flow, which can be rewritten as(54)∂Ns∂t+∇⋅(Nsvs)=-Rdiss.
Here the relaxation (dissipation) term, in accordance with (35), is equal to(55)Rdiss=2NsτGL[ξGL28mℏ2(mvs22+μs1)+NsNs°-1],
with τGL and ξGL being phenomenological parameters of the GL theory [17–21, 27] (see formulae (30)).If the right-hand side of (54) was absent, the universal solenoidality condition ∇·vs=0 for the superfluid flow would imply that the superfluid behaves as an incompressible ideal fluid. However, in effect this is not the case because from (54) it follows that (56)dNsdt≡(∂∂t+vs⋅∇)Ns=-Rdiss≠0,
where the right-hand side is not equal to zero in the general case. This refutes the conclusion about the superfluid incompressibility, which can incorrectly be inferred from the condition ∇·vs=0, as it was done before [26].It is convenient to representℛdiss in the following form:(57)Rdiss=ΓRNswithΓR≡1τR=2GτGL,
where, according to formula (35), we have (58)G=ξGL28mℏ2(mvs22+μs1)+NsNs°-1≡ξGL2(2mvsℏ)2-ξGL2∇2Ns1/2Ns1/2+NsNs°-1.
The last equality in (58) has been obtained by employing expression (12) for the chemical potential μs1.The quantityΓR≡τR-1=2G/τGL, appearing in (57) and playing a role of the effective pair-breaking rate, consists of two terms: ΓR=Γ1+Γ2, which correspond to the contributions from two terms in (58).Thefirst termΓ1 is related to the contribution of mvs2/2 to the quantity G and realized in the particular case of spatial uniformity, when Ns≃Ns° and the quantity μs1∝∇2Ns1/2 is vanishingly small, so that (59)Γ1=ξGL2τGL8mℏ2mvs2=2DGL(2mvsℏ)2,
where DGL=ξGL2/τGL.In the literature (e.g., see [21, 37, 38]), there is a similar quantity Γ=2D(mvs/ℏ)2 involving the normal-state diffusion constant D=vF2τ/3. If the diffusion constants D and DGL are identified with each other, the quantity Γ becomes four times smaller than Γ1 because it includes the electron mass m instead of the effective mass ms=2m for Cooper's pairs, as it occurs in the real case. It is known that Γ=τE-1/2+τs-1 near Tc [21, 37], whence(60)Γ1=4Γ=2(1τE+2τs).HereτE is the energy relaxation time due to the inelastic electron-phonon collisions and τs is the elastic spin-flip scattering time due to magnetic impurities (if any), with both being taken for the normal state of a superconductor.Thesecond termΓ2 is related to the contributions from μs1 and the two last terms in (58) to the quantity G and caused by spatial nonuniformity of Ns≠Ns°. This is also due to the pair-breaking processes with participating phonons, whose relaxation time (τph≈τE) is usually much more than the GL relaxation time τGL [18, 21, 38].So in order to ensureτE≈τR≡τGL/2G≫τGL, it is necessary that values of G should be sufficiently small. The above inequality provides sufficiently fast formation of the quasilocal equilibrium in the condensate (characterized by τGL) as compared with the relatively slow pair-breaking processes (characterized by τE≈τR). Such an assertion should be true at any temperature near Tc when τGL≃πℏ/8kB(Tc-T) [18, 21, 38]. Consequently, the closer temperature T is to Tc, the smaller should be the quantity G (G→0 in the limiting case T→Tc) to ensure the inequality τGL≪τE≈τR≡τGL/2G. This provides fast formation of the quasi-local equilibrium in the condensate under nonstationary physical conditions.Before rearranging the continuity equation (54), let us consider the extremely limiting case of G=0 and show that it can be realized, irrespective of temperature, only in the particular case of stationary situations described by the usual (time-independent) GL equation (20).


### 3.1. Stationary Ginzburg-Landau Equation

A comparison of equations (20) and (29) displays a condition for realizing the GL equation in the form of γ=0 imposed on (29), which reflects the dominant effect of relaxation processes. It is clearly evident from (31) that the condition γ=0 (or τGL=0) is equivalent to the equation R̂Ψ=0, where the relaxation operator R̂ is defined as (32). From expression (34), it follows that the stationary GL equation (20) assumes the form(61)R̂Ψ≡-(G-iξGL22mℏ∇⋅Jsρs)Ψ=0.The quantity (61) must be equal to zero for any arbitrary function Ψ, which is fulfilled if and only if both the real and imaginary terms enclosed in parentheses vanish.Vanishing thereal part of R̂ in (61) means G=0. Formula (58) for G, being written in terms of the normalized modulus ψ=|Ψ|/|Ψ°|≡(Ns/Ns°)1/2 of the complex order parameter Ψ, converts the equality G=0 into the following equation:(62)ξGL2∇2ψ+[1-ξGL2(2mvsℏ)2]ψ-ψ3=0
(ψ here and below should not be confused with the microscopic wave function ψ mentioned in Section 1).The general expressionvs=-(e/m)A for the simply connected and vortex-free superconductors [17, 18, 25] allows one to write the following relation:(63)2mvsℏ=2eAℏ≡2πAϕ0,
with ϕ0=πℏ/e being the quantum of magnetic flux (written in the SI system of units). Then equation (62) assumes the following form:(64)ξGL2∇2ψ+[1-ξGL2(2eAℏ)2]ψ-ψ3=0.For simply-connected and vortex-free superconductors the order parameterΨ is always real-valued, then ψ=Ψ/Ψ°≡(Ns/Ns°)1/2. Consequently, (64) is the ordinary GL equation and it fully coincides with the similar ones existing now in the literature (e.g., see books [17, 18, 25]).Vanishing theimaginary part of R̂ in (61) establishes the following requirement for stationary situations:(65)∇⋅Js=0.The requirement (65) removes the chemical potential μs2 of relaxation nature from both the chemical potential relation (37) and the force equation (52) because μs2∝∇·Js=0, in accordance with (38).Therefore, in the stationary situation with steady-state superfluid flow described by the GL equation (62) or (64), the superfluid motion is governed by the simplified equation (17), which is due to μs2=0, rather than the general equation (52). In (17), there are the uniform chemical potential μs0 given by (49) and the nonuniform chemical potential δμs=μs1 determined by the quantum potential Qs of Cooper's pairs in the form of (18).Since the superfluid flow is always solenoidal, the requirement (65) by virtue of the solenoidality condition ∇·vs=0 (see formula (44)) yields(66)vs⋅∇Ns=0.So under stationary conditions we have∇Ns⊥vs, that is, any spatial nonuniformity in Ns arises only in the plane perpendicular to the supervelocity vector vs. Thus, the longitudinal component of the gradient ∇Ns (∇Ns∥vs) can appear solely in nonstationary situations when the requirement (65) and its consequence (66) fail to occur for solenoidal motion of the superfluid component.


### 3.2. Time-Dependent Ginzburg-Landau Equation

In the case of nonstationary situations, the general continuity equation (54) for the superfluid flow can be rearranged with the help of (57) and (58) and reduced to the following form:(67)DGL∇2ψ-dψdt=-[(1-ψ2)-ξGL2(2mvsℏ)2]ψτGL,
where DGL=ξGL2/τGL and dψ/dt=∂ψ/∂t+vs·∇ψ.Equation (67) is the desired version of the time-dependent Ginzburg-Landau equation which describes the time evolution of the normalized modulus ψ=|Ψ|/|Ψ°|≡(Ns/Ns°)1/2 of the nonequilibrium order parameter Ψ(r,t).Expressions (59) and (60) with τs-1=0 (no magnetic impurities) yield a ratio of the relaxation times (68)τGLτE=ξGL2(2mvsℏ)2=ξGL2(2eAℏ)2≪1.
Here the second equality conforms to simply connected and vortex-free superconductors, whereas the inequality physically means relatively fast formation of the quasi-local equilibrium (as compared with the pair-breaking processes) in the condensate under nonstationary conditions.Application of formula (68) reduces the general TDGL equation (67) to the simplified form (69)DGL∇2ψ-dψdt=-(1-ψ2)ψτGL,
where we have used the inequality τGL≪τE.It is evident that the general form (67) of the TDGL equation assumes the special form (62) of the GL equation with τGL≪τE if dψ/dt=0. This is indeed realized under the stationary conditions when (i) ∂ψ/∂t=0 and (ii) vs·∇ψ≡(4NsNs°)-1/2(vs·∇Ns)=0 as a result of the stationary requirement (66). In this case, (69) also changes its form into the known one [17, 18, 25](70)ξGL2∇2ψ+ψ-ψ3=0,
which fully coincides with the stationary GL equations (62) and (64) when employing the inequality (68).Thus, any time-dependent and space-nonuniform physical situation is governed by the equation of superfluid motion in hydrodynamic form of (52) and the TDGL equation (67) for the modulus of the order parameter. The superfluid motion equation (52) follows from the chemical potential relation (37) and is fully equivalent to this relation. Our version (67) of the TDGL equation, being a consequence of the continuity equation (54), is of charge conservation character. In general, this equation has the form of a nonhomogeneous diffusion equation with nonlinear terms allowing for the superfluid motion and dissipation processes, which assumes the conventional form (62) of the GL equation under stationary conditions.


## 3.1. Stationary Ginzburg-Landau Equation

A comparison of equations (20) and (29) displays a condition for realizing the GL equation in the form of γ=0 imposed on (29), which reflects the dominant effect of relaxation processes. It is clearly evident from (31) that the condition γ=0 (or τGL=0) is equivalent to the equation R̂Ψ=0, where the relaxation operator R̂ is defined as (32). From expression (34), it follows that the stationary GL equation (20) assumes the form(61)R̂Ψ≡-(G-iξGL22mℏ∇⋅Jsρs)Ψ=0.The quantity (61) must be equal to zero for any arbitrary function Ψ, which is fulfilled if and only if both the real and imaginary terms enclosed in parentheses vanish.Vanishing thereal part of R̂ in (61) means G=0. Formula (58) for G, being written in terms of the normalized modulus ψ=|Ψ|/|Ψ°|≡(Ns/Ns°)1/2 of the complex order parameter Ψ, converts the equality G=0 into the following equation:(62)ξGL2∇2ψ+[1-ξGL2(2mvsℏ)2]ψ-ψ3=0
(ψ here and below should not be confused with the microscopic wave function ψ mentioned in Section 1).The general expressionvs=-(e/m)A for the simply connected and vortex-free superconductors [17, 18, 25] allows one to write the following relation:(63)2mvsℏ=2eAℏ≡2πAϕ0,
with ϕ0=πℏ/e being the quantum of magnetic flux (written in the SI system of units). Then equation (62) assumes the following form:(64)ξGL2∇2ψ+[1-ξGL2(2eAℏ)2]ψ-ψ3=0.For simply-connected and vortex-free superconductors the order parameterΨ is always real-valued, then ψ=Ψ/Ψ°≡(Ns/Ns°)1/2. Consequently, (64) is the ordinary GL equation and it fully coincides with the similar ones existing now in the literature (e.g., see books [17, 18, 25]).Vanishing theimaginary part of R̂ in (61) establishes the following requirement for stationary situations:(65)∇⋅Js=0.The requirement (65) removes the chemical potential μs2 of relaxation nature from both the chemical potential relation (37) and the force equation (52) because μs2∝∇·Js=0, in accordance with (38).Therefore, in the stationary situation with steady-state superfluid flow described by the GL equation (62) or (64), the superfluid motion is governed by the simplified equation (17), which is due to μs2=0, rather than the general equation (52). In (17), there are the uniform chemical potential μs0 given by (49) and the nonuniform chemical potential δμs=μs1 determined by the quantum potential Qs of Cooper's pairs in the form of (18).Since the superfluid flow is always solenoidal, the requirement (65) by virtue of the solenoidality condition ∇·vs=0 (see formula (44)) yields(66)vs⋅∇Ns=0.So under stationary conditions we have∇Ns⊥vs, that is, any spatial nonuniformity in Ns arises only in the plane perpendicular to the supervelocity vector vs. Thus, the longitudinal component of the gradient ∇Ns (∇Ns∥vs) can appear solely in nonstationary situations when the requirement (65) and its consequence (66) fail to occur for solenoidal motion of the superfluid component.


## 3.2. Time-Dependent Ginzburg-Landau Equation

In the case of nonstationary situations, the general continuity equation (54) for the superfluid flow can be rearranged with the help of (57) and (58) and reduced to the following form:(67)DGL∇2ψ-dψdt=-[(1-ψ2)-ξGL2(2mvsℏ)2]ψτGL,
where DGL=ξGL2/τGL and dψ/dt=∂ψ/∂t+vs·∇ψ.Equation (67) is the desired version of the time-dependent Ginzburg-Landau equation which describes the time evolution of the normalized modulus ψ=|Ψ|/|Ψ°|≡(Ns/Ns°)1/2 of the nonequilibrium order parameter Ψ(r,t).Expressions (59) and (60) with τs-1=0 (no magnetic impurities) yield a ratio of the relaxation times (68)τGLτE=ξGL2(2mvsℏ)2=ξGL2(2eAℏ)2≪1.
Here the second equality conforms to simply connected and vortex-free superconductors, whereas the inequality physically means relatively fast formation of the quasi-local equilibrium (as compared with the pair-breaking processes) in the condensate under nonstationary conditions.Application of formula (68) reduces the general TDGL equation (67) to the simplified form (69)DGL∇2ψ-dψdt=-(1-ψ2)ψτGL,
where we have used the inequality τGL≪τE.It is evident that the general form (67) of the TDGL equation assumes the special form (62) of the GL equation with τGL≪τE if dψ/dt=0. This is indeed realized under the stationary conditions when (i) ∂ψ/∂t=0 and (ii) vs·∇ψ≡(4NsNs°)-1/2(vs·∇Ns)=0 as a result of the stationary requirement (66). In this case, (69) also changes its form into the known one [17, 18, 25](70)ξGL2∇2ψ+ψ-ψ3=0,
which fully coincides with the stationary GL equations (62) and (64) when employing the inequality (68).Thus, any time-dependent and space-nonuniform physical situation is governed by the equation of superfluid motion in hydrodynamic form of (52) and the TDGL equation (67) for the modulus of the order parameter. The superfluid motion equation (52) follows from the chemical potential relation (37) and is fully equivalent to this relation. Our version (67) of the TDGL equation, being a consequence of the continuity equation (54), is of charge conservation character. In general, this equation has the form of a nonhomogeneous diffusion equation with nonlinear terms allowing for the superfluid motion and dissipation processes, which assumes the conventional form (62) of the GL equation under stationary conditions.


## 4. Conclusion

The macroscopic equations of superfluid transport have been derived on the basis of a phenomenological combination of the usual Ginzburg–Landau equation and Schrödinger's equation for the macroscopic wave function (see (24)–(28)), which has resulted in the desired form of (31) to describe nonstationary superconductivity. Application of the Madelung-Feynman approach [8, 22, 23] to the newly derived equation (31) by separating it into the real and imaginary parts has finally provided us with the required equation (52) to describe the superfluid motion in hydrodynamic form. Besides, such an approach has given: (i) four different contributions to the total chemical potential μs for superfluid component in the form of (39) and (ii) the continuity equation (36) or (54) for the superfluid flow with the relaxation term in the form of (55) involving the phenomenological Ginzburg-Landau parameters τGL and ξGL.The hydrodynamic equation (52) of superfluid motion and the novel version (67) of the TDGL equation for the modulus of the order parameter have turned out to be a consequence, respectively, of the chemical potential relation (37) and of the continuity equation (54). This result explains why, unlike semiconductor electronics, in stationary superconductor theory nobody applies the equation of superfluid motion and the continuity equation. Instead of them, the chemical potential relation (37) and the GL equation (62) are commonly used.Taking into account quantum dissipation effects with the help of the relaxation term-λv entering into the hydrodynamic equation of motion in the form of (2), as is commonly used for nonsuperconductive media, is not the case for the theory of time-dependent superconductivity. For the latter, the hydrodynamic equation (52) of superfluid motion has taken into account the relaxation processes in terms of μs2 given by (38). Again, the dissipative effects have proved to be inserted into the continuity equation (54) or into its output in the form of the time-dependent Ginzburg-Landau equation (67) for the modulus of the order parameter.All the above results are mutually consistent within the scope of the chosen macroscopic description and, being phenomenological and model-neutral, can be applied to both the low-Tc and high-Tc superconductors.



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