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Linear-in-Temperature Resistivity and Planckian Dissipation Arise in a Stochastic Quantization Model of Cooper Pairs

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03 March 2024

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04 March 2024

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Abstract
We suppose that a Cooper pair (CP) will experience a damping force exerted by the condensed matter. A Langevin equation of a CP in two dimensional condensed matter is established. Following a method similar to Nelson's stochastic mechanics, generalized Schr\"{o}dinger equation of a CP in condensed matter is derived. If the CPs move with a constant velocity, then the corresponding direct current (DC) electrical conductivity can be calculated. Therefore, a Drude like formula of resistivity of CPs is derived. We suppose that the damping coefficient of CPs in two dimensional cuprate superconductors is a linear function of temperature. Then the resistivity and scattering rate of CPs turn out to be also linear-in-temperature. The origin of linear-in-temperature resistivity and Planckian dissipation in cuprate superconductors may be the linear temperature dependence of the damping coefficient of CPs.
Keywords: 
Planckian resistivity; Planckian dissipation; Cooper pair; strange metal; stochastic mechanics
Subject: 
Physical Sciences  -   Condensed Matter Physics

1. Introduction

It is known that the resistivity ρ of the normal states of cuprate superconductors obeys the following relationship [1,2,3,4]
ρ = ρ 0 + A T ,
where ρ 0 is the residual resistivity, A is a coefficient independent of temperature, T is temperature.
These phenomena of linear temperature dependence of resistivity are found in numerous strongly correlated electron systems, such as the heavy fermion compounds [1,3], transition metal oxides [1,4,5], iron pnictides [1], magic angle twisted bilayer graphene, organic metals [1] and conventional metals [1], often in connection with unconventional superconductivity. Sometimes this linear-in-temperature resistivity is called Planckian resistivity [2]. When superconductivity is destroyed by a high magnetic field, the recovered normal state still obeys this law of linear-in-temperature resistivity in the low temperature region [4]. In most of the heavy fermion materials, the linear-in-temperature resistivity appears when they have been tuned by some external parameter to create a low-temperature continuous phase transition which is referred to a quantum critical point (QCP) [1]. Thus, the linear temperature dependence of resistivity are often associated with quantum criticality. The linear-in-temperature resistivity of L S C O with different gradients, different doping dependencies and different origins appears not only at high temperature but also at low temperature [6].
Strange metal behavior refers to a linear temperature dependence of the electrical resistivity [1,3]. A unified theory of this scaling law (1) in different strange metals is still an open problem [2,3].
Before the discovery of quantum mechanics, a successful formula of resistivity of metals is proposed in the Drude model ([7], p. 7). Shortly after the discovery of quantum mechanics, Sommerfeld improved the Drude model. In the Sommerfeld model, the following Drude formula of resistivity of metals can be derived approximately based on quantum theoty ([7], p. 251)
ρ = m * n e 2 τ ,
where n is the number dendity of electrons, e is the electric charge of an electron, m * is the effective mass of an electron, τ is the relaxation time of an electron.
If the transport scattering rate 1 / τ is linear-in-temperature and is the only temperature-dependent quantity in Eq.(2), then the scaling law (1) of resistivity can be derived directly. Thus, a clue to study the scaling law (1) is to investigate the relaxation time τ in the Drude formula (2).
The Drude formula (2) is valid only for charge carriers which obey the Fermi-Dirac distribution. Experiments have shown that the dominant charge carriers in Y B a 2 C u 3 O 7 δ (YBCO) film are Cooper pairs (CPs) [8]. Since CPs are not Fermions, the Drude formula (2) may be not valid in the normal states of cuprate superconductors. Thus, an interesting question is that whether a similar formula for the resistivity of the normal states of cuprate superconductors exists.
According to the Heisenberg uncertainty principle, a local equilibration time of any many-body quantum system cannot be faster than the following Planckian time τ p [3]
τ p = k B T ,
where h is the Plank constant, = h / 2 π , k B is the Boltzmann constant.
This timescale τ p is associated with quantum criticality and known to bound the validity of a Boltzmann description of transport [9]. τ p is suggested to be the lower bound of the phase coherence time in quantum critical systems [9]. τ p is also known to control the electronic dynamics of the cuprate strange metal [9]. Thus, an idea is that the relaxation time τ of CP in cuprate superconductors may be proportional to the Planckian time τ p , i.e., τ = α 0 τ p , where α 0 is a dimensionless parameter. Indeed, experiments have shown that the scattering rate 1 / τ in the region of the temperature-linear resistivity of a wide range of metals, including heavy fermion, oxide [4,5,8], pnictide, organic metals and conventional metals, can be written as [1]
1 τ = α 0 k B T ,
where α 0 1 .
Eq.(4) shows that the relaxation time τ is approximately equal to the Planckian time τ p , i.e., τ τ p . The case of α 0 1 is referred to the Planckian dissipation [3]. It is surprising that the linear-in-temperature scattering rate 1 / τ and the behaviors of Planckian dissipation in these materials (except the conventional metals) can be seen down to low temperatures with appropriate tuning by magnetic field, chemical composition or hydrostatic pressure [1]. It is suggested that there may be a fundamental principle governing the transport of CPs [8].
If Eq.(4) and Eq.(2) are valid in the normal states of cuprate superconductors, then Eq.(1) may be derived. In this manuscript we focus on this clue and try to derive the scaling law (1).

2. Stochastic Mechanics of a Cooper Pair in Two Dimensional Condensed Matter

In order to explain the energy quantization of atoms, E. Schrödinger proposes the following equation for a non-relativistic particle moving in a potential [10]
i ψ t = 2 2 m 2 ψ + U ( r ) ψ ,
where t is time, r is a point in space, ψ ( r , t ) is the wave function, m is the mass of the particle, U ( r ) is the potential, h is the Plank constant, = h / 2 π , 2 2 / r 1 2 + 2 / r 2 2 + 2 / r 3 2 is the Laplace operator in a Cartesian coordinate { r 1 , r 2 , r 3 } .
The Schrödinger equation (5) is a fundamental assumption in non-relativistic quantum mechanics [10]. Although the Schrödinger equation can be used to describe some non-relativistic quantum phenomena, the origin of quantum phenomena remains an unsolved problem in physics for more than 100 years [11,12]. Although the axiomatic system of quantum mechanics was firmly established, the interpretation of quantum mechanics is still open [11,12]. There exist some paradoxes in quantum mechanics [13,14,15,16], for instance, the paradox of reduction of a wave packet and the paradox of the Schrödinger cat.
Fényes proposed an interpretation of quantum mechanics based on a Markov process. Fényes’ work was developed by Weizel and discussed by Kershaw [17]. According to Luis de Broglie [18], the success of the probabilistic interpretation of | ψ | 2 inspired Einstein to speculate that the probability | ψ | 2 is generated by a kind of hidden Brownian motions of particles. This kind of hidden motions was called quasi-Brownian motions by Luis de Broglie [18].
If the quantum phenomena stem from the stochastic motions of particles, then we may establish a more fundamental and more powerful theory of quantum phenomena other than quantum mechanics. The Schrödinger equation may no longer be a basic assumption and may be derived in this new theory. Indeed, E. Nelson [19] derived the Schrödinger equation by means of theory of stochastic processes based on the assumption that every particle with mass m in vacuum is subject to Brownian motion with diffusion constant / 2 m .
Inspired by Nelson’s stochastic mechanics [19,20,21,22,23,24,25,26,27,28], we propose a theoretical derivation of the Schrödinger equation based on Newton’s second law and a mechanical model of vacuum [29].
Recently, monolayer crystals of the high-temperature superconductor B i 2 S r 2 C a C u 2 O 8 + δ (Bi-2212) was obtained by a fabrication process [30]. The superconductivity, the pseudogap, charge order and the Mott state at various doping concentrations of the monolayer Bi-2212 reveals that the phases are indistinguishable from those in the bulk [30]. Monolayer Bi-2212 displays the fundamental physics of cuprate superconductors [30]. Therefore, monolayer copper oxides is a platform for studying high-temperature superconductivity in two dimensions. Thus, we focus on two dimensional condensed matters.
Modern experiments, for instance, the Casimir effect [31,32], have shown that vacuum is not empty. Thus, we suppose that there is a damping force exerted on each particle by vacuum [29]. For a microscopic particle moving in vacuum, we have the following relation [29]
= 2 k 0 T 0 η 0 ,
where k 0 is a constant similar to the Boltzmann constant k B , T 0 is the temperature of the Ω ( 0 ) substratum in the vicinity of the particle in vacuum [33], η 0 is a damping coefficient related to vacuum.
It is known that a CP in condensed matter may be scattered by ions, electrons, phonon, etc. In the Drude theory of metals, the effect of individual electron collisions is approximately treated by introducing a damping force into the equation of motion of an electron ([7], p. 11). Following the Drude theory, we suppose that a CP in a condensed matter will experiences not only a damping force exerted by vacuum but also an additional damping force exerted by the condensed matter. We introduce a two dimensional Cartesian coordinate system { r 1 , r 2 } which is attached to the condensed matter. We suppose that the two dimensional velocity v = d r / d t of the CP exists. Applying Newton’s second law, the motion of a CP may be described by the following Langevin equation [34]
m c d 2 r d t 2 = η 0 m c v η 1 m c v η 2 m c d 2 r d t 2 + F ( r , t ) + ξ ( t ) ,
where m c is the mass of the CP, η 1 is a damping coefficient related to the condensed matter, η 2 is a quasi-inertial force coefficient, ξ ( t ) is a two dimensional random force and F ( r , t ) is a two dimensional external force field.
We introduce the following definitions
η 0 m c v η 1 m c v = η 0 m d v ,
m c d 2 r d t 2 + η 2 m c d 2 r d t 2 = m q d 2 r d t 2 ,
where m d is the damping mass of the CP, m q is the quasi-inertial mass of the CP.
Eq.(8) and Eq.(9) can be written as
m d = η 0 + η 1 η 0 m c ,
m q = ( 1 + η 2 ) m c ,
Using Eq.(8) and Eq.(9), Eq.(7) can be written as
m q d 2 r d t 2 = η 0 m d v + F ( r , t ) + ξ ( t ) .
Let ξ i ( t ) be the ith component of the random force ξ ( t ) , i.e., ξ ( t ) = ( ξ 1 ( t ) , ξ 2 ( t ) ) .
Assumption 1. 
Assume that the force field F ( r , t ) is a continuous function of r and t. Inspired by the Ornstein-Uhlenbeck theory [35,36] of Brownian motion, we suppose that the random force ξ exerted on the CP by the condensed matter is a two-dimensional Gaussian white noise [37,38,39,40] and the variance E ( ξ i 2 ) of the ith component of ξ is [29]
E ( ξ i 2 ( t ) ) σ i 2 = 2 η 0 m d k 0 T ω ,
where σ i > 0 , i = 1 , 2 , k 0 is a parameter similar to the Boltzmann constant, T ω is the temperature of the Ω ( 0 ) substratum [33] in the location of the condensed matter.
For convenience, we introduce the following notation
D 1 = σ 1 2 2 = η 0 m d k 0 T ω .
The Gaussian white noise ξ is the generalized derivative of a Wiener process Q ( t ) [37,39]. We can write formally [37,38,39,40]
ξ ( t ) = d Q ( t ) d t ,
where Q ( t ) is a two-dimensional Wiener process with a diffusion constant D 1 .
The mathematically rigorous form of Eq.(12) is the following stochastic differential equations based on Itô stochastic integral [36]
d r ( t ) = v ( t ) d t , m q d v ( t ) = η 0 m d v ( t ) d t + F ( r , t ) d t + d Q ( t ) ,
where r ( t ) , v ( t ) , F ( r , t ) and Q ( t ) , t 0 are stochastic processes on a probability space ( Ω , F , P ) , d Q ( t ) are independent of all of the r ( s ) , v ( s ) , with s t , r ( 0 ) = r 0 , v ( 0 ) = v 0 .
The microstate of the CP at time t is defined by the random vector ( r ( t ) , v ( t ) ) [29].
Proposition 1. 
Suppose that Eqs.(13) are valid and the force field F ( r , t ) : R 2 × R + R 2 satisfy a global Lipschitz condition, that is, for some constant C 0 ,
| F ( r 1 , t ) F ( r 2 , t ) | C 0 | r 1 r 2 | ,
for all r 1 and r 2 in R 2 , where R 2 denotes the two-dimensional Descartes space, R + denotes the set of positive real numbers. Then, at a time scale of an observer very large compare to the relaxation time τ c ,
τ c m q η 0 m d ,
the solution r ( t ) of the Langevin equation Eq.(16) converges to the solution y ( t ) of the following Smoluchowski equation Eq.(20) with probability one uniformly for t in compact subintervals of [ 0 , ) for all v 0 , i.e.,
lim 1 / τ c r ( t ) = y ( t ) ,
where y ( t ) is the solution of the following Smoluchowski equation
d y ( t ) = b ( y , t ) d t + d w ( t ) ,
where y ( 0 ) = r 0 , w ( t ) is a two-dimensional Wiener process with a diffusion constant D 3 defined by
D 3 = k 0 T ω η 0 m d .
A proof of Proposition 1 can be found in the Appendix A. Following similar methods in Ref. [29], a Schrödinger like equation (98) and Eq.(64) can be derived, refers to Appendix B.
Putting Eq.(64) into Eq.(98), we have the following result.
Proposition 2. 
The Schrödinger like equation Eq.(98) reduces to the following Schrödinger like equation
i ψ t = 2 2 m w 2 ψ + U ( r ) ψ ,
where m w is the wave mass defined by Eq.(63).
From Eq.(10), in vacuum the damping mass m d reduces to the mass m c of a CP. In vacuum, T ω reduces to T 0 . Therefore, in vacuum the wave mass m w defined by Eq.(63) reduces to m c . Thus, the Schrödinger like equation (22) in the condensed matter is a generalization of the Schrödinger equation (5) in vacuum.

3. Calculation of Direct Current (DC) Electrical Conductivity

If there is no external magnetic field and the external electric field E is a constant vector field, then the Langevin equation (12) can be written as
m q d 2 r d t 2 = η 0 m d v + e c E + ξ ( t ) ,
where e c is the electric charge of the CP.
If the mean velocity v of the CP is high enough such that d v / d t = 0 , then we call this velocity as drift velocity and denotes it as v d . Thus, if the observer look at the CP for a time long enough comparing to the relaxation time τ c , then he will observe the long time averaged quantities of the Langevin equation (23). Since ( d v / d t ) | v = v d = 0 and E ξ ( t ) = 0 , the long time averaged form of the Langevin equation (23) can be written as ([7], p. 7; [41], p. 16)
v d = e c E η 0 m d .
The current density j corresponding to the drift velocity v d is ([7], p. 7; [41], p. 16)
j = n c e c v d ,
where n c is the number density of CPs.
Putting Eq.(24) and Eq.(25), we have
j = n c e c 2 η 0 m d E .
Eq.(25) can be written as ([7], p. 7)
j i = j σ i j E j ,
where j i is the ith component of the current density j , E j is the jth component of the electric field E , σ i j is the conductivity tensor which can be written as
σ i j = σ δ i j ,
where
σ = n c e c 2 η 0 m d ,
δ i j is the Kronecker symbol
It is known that the resistivity of the normal states of cuprate superconductors exhibits strong anisotropy ([42], p. 190). Thus, Eq.(28) may be only valid for the plane conductivity ρ a b of two dimensional cuprate superconductors and not valid for bulk cuprate. Noticing ρ a b = 1 / σ and Eq.(29), we have
ρ a b = η 0 m d n c e c 2 .
Using Eq.(18), Eq.(30) can also be written as
ρ a b = m q n c e c 2 τ c .

4. Linear Temperature Dependence of Resistivity in the Normal States of Cuprate Superconductors

Using Eq.(10), Eq.(30) can be written as
ρ a b = ( η 0 + η 1 ) m c n c e c 2 .
Inspired by Eq.(32), we speculate that the origin of the linear-in-temperature resistivity of the strange metals may be the linear temperature dependence of the damping coefficient η 0 + η 1 . Thus, we introduce the following assumption.
Assumption 2. 
Suppose that the following relationship is valid in the strange metal states of two dimensional cuprate superconductors
η 0 + η 1 η 0 = b 0 T T 0 .
where b 0 ia a parameter to be determined.
In vacuum we have T = T 0 and η 1 = 0 [29]. Suppose that Eq.(33) is also valid in vacuum. Thus, we have b 0 = 1 . Using Eq.(33), Eq.(32) can be written as
ρ a b = A 1 T ,
where
A 1 = η 0 m c n c e c 2 T 0 .
Noticing Eq.(6), Eq.(34) can also be written as
ρ a b = A 2 T ,
where
A 2 = 2 k 0 m c n c e c 2 .
If A 2 is independent of temperature T, then Eq.(34) shows that the plane resistivity ρ a b is a linear function of temperature T. Thus, Eq.(1) is derived based on the stochastic quantization model of CP in two dimensional cuprate superconductors.
A prediction of Eq.(36) is that if T = 0 , then ρ a b = 0 , i.e., the residual resistivity ρ 0 = 0 .

5. Linear Temperature Dependence of Scattering Rate of CPs

Using Eq.(10) and Eq.(11), Eq.(18) can be written as
1 τ c = η 0 + η 1 1 + η 2 .
According to Eq.(38), the origin of the linear-in-temperature scattering rate 1 / τ c of the cuprate strange metals may be the linear temperature dependence of the damping coefficient η 0 + η 1 . Noticing Eq.(33), Eq.(38) can be written as
1 τ c = c 1 T ,
where
c 1 = η 0 T 0 ( 1 + η 2 ) .
Noticing Eq.(6), Eq.(39) can also be written as
1 τ c = c 2 k B T ,
where
c 2 = 2 k 0 k B ( 1 + η 2 ) .
If c 1 is independent of temperature T, then Eq.(39) shows that the scattering rate 1 / τ c is a linear function of temperature T.
If we suppose that c 2 1 , then Eq.(4) is derived. Thus, we may say that CPs in the cuprate strange metals are undertaking the Planckian dissipation [3].

6. Conclusion

The origin of the linear-in-temperature resistivity of the normal state of hole-doped cuprate superconductors is a unsolved problem. Inspired by the Drude formula of resistivity, we speculate that the transport scattering rate of CPs in the normal states of cuprate superconductors may be linear-in-temperature. Thus, a clue to explain the linear-in-temperature scaling law of resistivity in strange metal states of cuprate superconductors is to seek a Drude like formula of resistivity and investigate the relaxation time of CP dynamics. We suppose that a CP in a condensed matter will experiences not only a damping force exerted by vacuum but also an additional damping force exerted by the condensed matter. Thus, a Langevin equation of a CP in two dimensional condensed matter is established. Following a similar method of Nelson’s stochastic mechanics, generalized Schrödinger equation in condensed matter is derived. If CPs move with a constant velocity, then the electrical current density corresponding to the drift velocity can be calculated. Therefore, a Drude like formula of resistivity of CPs is derived. The damping coefficient of CPs in two dimensional cuprate superconductors is supposed to be a linear function of temperature. Thus, the plane resistivity and scattering rate of CPs turn out to be also linear functions of temperature.

Appendix A: Proof of Proposition 1

We introduce the following definitions
β = η 0 m d m q ,
K ( r , t ) = F ( r , t ) m q ,
B ( t ) = Q ( t ) m q .
Then, B ( t ) is a two-dimensional Wiener process with a diffusion constant D 2 [36]
D 2 = D 1 m q 2 = η 0 m d k 0 T ω m q 2 = β k 0 T ω m q .
Using Eqs.(43-), Eq.(16) can be written as
d r ( t ) = v ( t ) d t , d v ( t ) = β v ( t ) d t + K ( r , t ) d t + d B ( t ) .
We introduce the following definitions
b ( r , t ) = K ( r , t ) β ,
w ( t ) = B ( t ) β .
Noticing Eq.(43), w ( t ) is a two-dimensional Wiener process with a diffusion constant D 3 [36]
D 3 = D 2 β 2 = k 0 T ω η 0 m d .
Using Eq.(48-), Eq.(47) can be written as
d r ( t ) = v ( t ) d t , d v ( t ) = β v ( t ) d t + β b ( r , t ) d t + β d w ( t ) .
Let r ( t ) be the solution of Eq.(51) with r ( 0 ) = r 0 , v ( 0 ) = v 0 . According to Eq.(17), the functions b ( r , t ) : R 2 × R + R 2 also satisfies a global Lipschitz condition. For a time scale of an observer very large compare to the relaxation time τ c 1 / β , he concludes that β can be regarded as infinity, i.e., β + . Applying Nelson’s Theorem 10.1 ([43], p. 59), the solution r ( t ) of the Langevin equation Eq.(51) converges to the solution y ( t ) of the Smoluchowski equation Eq.(20) with probability one uniformly for t in compact subintervals of [ 0 , ) for all v 0 . □

Appendix B: Generalized Schrödinger equation of a CP in two dimensional condensed matter

Noticing the asymmetry in time t, we can introduce the following Langevin equation [19]
d r ( t ) = v ( t ) d t , r ( 0 ) = r 0 m d v ( t ) = f v ( t ) d t + F ( r , t ) d t + d Q * ( t ) ,
where d N * ( t ) are independent of all of the r ( s ) , v ( s ) , with s t , v ( 0 ) = v 0 .
We define the following mean forward derivative D y ( t ) and the mean backward derivative D * y ( t ) [19]
D y ( t ) = lim t 0 + E t y ( t + t ) y ( t ) t ,
D * y ( t ) = lim t 0 + E t y ( t ) y ( t t ) t ,
where E t denotes the conditional expectation given the state of the system at time t.
We also have another Smoluchowski equation [19]
d y ( t ) = b * ( y , t ) d t + d w * ( t ) ,
where w * ( t ) has the same properties as w ( t ) except that the d w * ( t ) are independent of the y ( s ) with s t .
Based on Eq.(53-54), we have [19]
D y ( t ) = b ( y , t ) ,
D * y ( t ) = b * ( y , t ) .
For the probability density ρ ( y , t ) of y , we have the following forward Fokker-Planck equation and the backward Fokker-Planck equation [19]
ρ t = · ( ρ b ) + D 3 2 ρ ,
ρ t = · ( ρ b * ) D 3 2 ρ ,
where · / r 1 + / r 2 is the divergence operator in the two dimensional Cartesian coordinate { r 1 , r 2 } , 2 2 / r 1 2 + 2 / r 2 2 is the two dimensional Laplace operator.
We introduce the definitions of current velocity v 1 ( t ) and osmotic velocity u 1 ( t ) [19]
v 1 = 1 2 ( b + b * ) ,
u 1 = 1 2 ( b b * ) .
The current velocity v 1 ( t ) is the deterministic part of the total velocity b ( t ) of the CP. The osmotic velocity u 1 ( t ) is the stochastic part of the total velocity b ( t ) . The non-zero osmotic velocity u 1 ( t ) is a difference between stochastic mechanics deterministic mechanics [29].
We have the following result [19]:
u 1 = D 3 ρ ρ = D 3 ( ln ρ ) .
We introduce the following definition of wave mass.
Definition 1. 
The wave mass of the particle is defined by
m w T 0 T ω m d .
Using Eq.(6), Eq.(63) and Eq.(21), we have
D 3 = 2 m w .
Similar to the method of Ref. [19], we introduce the following definition of osmotic potential R 1
m w u 1 = R 1 ,
where the osmotic potential R 1 is defined by
R 1 m w D 3 ln ρ .
We introduce the definition of the mean second derivative a ( t ) of the stochastic process y ( t ) [19]
a ( t ) = 1 2 D D * y ( t ) + 1 2 D * D y ( t ) .
Applying a similar method of E. Nelson [19], we can derive the following Proposition 3 [29].
Proposition 3. 
The current velocity field v 1 ( r , t ) and the osmotic velocity field u 1 ( r , t ) satisfy the following coupled equations:
v 1 t = F m w ( v 1 · ) v 1 + ( u 1 · ) u 1
+ D 3 2 u 1 ,
u 1 t = D 3 ( · v 1 ) ( v 1 · u 1 ) .
Similar to the deterministic newtonian mechanics, we can also introduce the following concept of deterministic momentum field p d ( r , t ) and stochastic momentum field p s ( r , t ) of the Brownian particle:
p d ( r , t ) = m w v 1 ( r , t ) ,
p s ( r , t ) = m w u 1 ( r , t ) .
Proposition 4. 
If there exists a functions S 1 ( r , t ) such that
p d = S 1 ,
then, the deterministic momentum field p d ( r , t ) and stochastic momentum field p s ( r , t ) of the Brownian particle satisfy the following equations
p d ( t ) t = F 1 2 m w ( p d 2 ) + 1 2 m w ( p s 2 )
+ D 3 2 p s ,
p s ( t ) t = D 3 2 p d 1 m w ( p d · p s ) .
Proof of Proposition 4. We have the following equations in field theory:
( a · b ) = ( a · ) b + ( b · ) a + a × ( × b ) + b × ( × a ) ,
2 a = ( · a ) × ( × a ) ,
× ( φ ) = 0 ,
where a and b are arbitrary vectors, φ is an arbitrary scalar function.
Using Eq.(65), Eq.(72), Eq.(75) and Eq.(77), we have
1 2 ( v 1 2 ) = ( v 1 · ) v 1 ,
1 2 ( u 1 2 ) = ( u 1 · ) u 1 .
Using Eq.(72), Eq.(76) and Eq.(77), we have
2 v 1 = ( · v 1 ) .
Putting Eq.(78-79) into Eq.(68) and using Eq.(65) and Eq.(72), we obtain Eq.(73). Putting Eq.(80) into Eq.(69) and using Eq.(65) and Eq.(72), we obtain Eq.(74). □
We may call the functions S 1 ( r , t ) defined in Eq.(72) as the current potential. The current potential S 1 ( r , t ) is not uniquely defined by the deterministic momentum field p d ( r , t ) . For instance, let S 1 = S 1 + c 0 , where c 0 is an arbitrary constant. Then, we also have ( S 1 ) = p d .
Theorem 1. 
If there exist two functions U ( r ) and S 1 such that
F ( r , t ) = U ( r ) ,
p d = S 1 ,
then, the generalized Hamilton’s principal function
W 1 S 1 i R 1
satisfies the following generalized Hamilton-Jacobi equation
W 1 t = 1 2 m w ( W 1 ) 2 + U ( r ) i D 3 2 W 1 + θ 1 ( t ) + i θ 2 ( t ) ,
where θ 1 ( t ) and θ 2 ( t ) are two unknown real functions of t.
Proof of Theorem 1. We multiply Eq.(73) with 1 and then plus Eq.(74) multiplied by i. Thus, we obtain
( p d i p s ) t = F + 1 2 m w [ ( p d i p s ) 2 ] i D 3 2 ( p d i p s ) .
We introduce the following definition
p p d i p s , i 2 = 1 .
Thus, Eq.(85) becomes
p t = F + 1 2 m w ( p 2 ) i D 3 2 p .
We may regard the function S 1 and R 1 as the deterministic part and stochastic part of a generalized Hamilton’s principal function S defined by
W 1 S 1 i R 1 , i 2 = 1 .
Putting Eq.(65) and Eq.(82) into Eq.(86) and using Eq.(88), we have
p = W 1 .
Putting Eq.(89) into Eq.(87), we obtain
( W 1 ) t = F + 1 2 m w [ ( W 1 ) 2 ] i D 3 2 ( W 1 ) .
Noticing F = U ( r ) , Eq.(90) becomes
( W 1 ) t = V + 1 2 m w [ ( W 1 ) 2 ] i D 3 2 ( W 1 ) .
Eq.(91) can be written as
W 1 t + U ( r ) + 1 2 m w W 1 2 i D 3 2 ( W 1 ) = 0 .
Integration of Eq.(92) gives
W 1 t = U ( r ) + 1 2 m w W 1 2 i D 3 2 ( W 1 ) + θ 1 ( t ) + i θ 2 ( t ) ,
where θ 1 ( t ) and θ 2 ( t ) are two unknown real functions of t. □
The generalized Hamilton’s principal function W 1 S 1 i R 1 is not uniquely defined by p d . The reason is that p d = S 1 . Thus, S 1 is not uniquely defined by p d .
Similar to Bohr’s correspondence principle, we may also introduce the following correspondence principle in stochastic mechanics.
Assumption 3. 
If the diffusion constant D 3 is small enough, i.e., D 3 0 , then, the generalized Hamilton-Jacobi equation Eq.(84) in stochastic mechanics becomes identical to the following Hamilton-Jacobi equation in classical mechanics [44]
W t = 1 2 m ( W ) 2 + U ( r ) ,
where W ( r , t ) is a real function called Hamilton’s principal function, U ( r ) is a potential.
Theorem 2. 
Suppose that the assumptions Eq.(81 - 82) are valid. Then, the generalized Hamilton’s principal function W 1 satisfies the following generalized Hamilton-Jacobi equation
W 1 t = 1 2 m w ( W 1 ) 2 + U ( r ) i D 3 2 W 1 .
Proof of Theorem 2 Let D 3 = 0 . Then, from Eq.(62), we have u 1 = 0 . Thus, from Eq.(71), we have p s = 0 . Then, from Eq.65, R 1 is a constant. Thus, Eq.(84) can be written as
S 1 t = 1 2 m w ( S 1 ) 2 + U ( r ) + θ 1 ( t ) + i θ 2 ( t ) ,
According to Assumption 3, Eq.(96) should be identical to the Hamilton-Jacobi equation Eq.(94). Thus, we obtain θ 1 ( t ) = 0 and θ 2 ( t ) = 0 . □
Similar to the Hamiltonian mechanics [44], we introduce the following definition of wave function
ψ ( r , t ) = exp i W 1 ( r , t ) 2 m w D 3 .
The generalized Hamilton’s principal function W 1 is not uniquely defined by the deterministic momentum field p d . Therefore, the wave function ψ ( r , t ) defined by Eq.(97) is not uniquely defined by p d .
Theorem 3. 
The wave function ψ ( r , t ) satisfies the following Schrödinger like equation
i ψ t = D 3 2 ψ + 1 2 m w D 3 U ( r ) ψ .
Eq.(98) is equivalent to the generalized Hamilton-Jacobi equation Eq.(95).
Proof of Theorem 3. From the definition Eq.(97), we have
W 1 ( r , t ) = 2 m w D 3 i ln ψ ( r , t ) .
Putting Eq.(99) into Eq.(95), we obtain a Schrödinger like equation Eq.(98). Conversely, putting Eq.(97) into Eq.(98), we obtain the generalized Hamilton-Jacobi equation Eq.(95). □

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