Evolution and Dynamics of Quantum Fluids

Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations, that is semiclassical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum ﬂuid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity ﬁeld as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential were externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed ﬂuids have the main application of this equation. Quantum ﬂuids show strange behaviour when compared to normal ﬂuids. It is also shown that quantum ﬂuid also have spins which has no classical analog.


I. INTRODUCTION
Quantum hydrodynamics is one of the new research fields in fluid dynamics. Low-temperature superfluids show quantum mechanical behaviour. Classical hydrodynamics is governed by a single equation known as Navier-Strokes equation 1 . Often Navier Strokes equation is solved either computationally or using certain assumptions and boundary conditions. A complete analytical problem of the Navier-Strokes equation is still a challenge and also a Millenium problem. Recent developments in Mathematics, showed that finite time blowup of 3 dimensional Navier-Strokes equation exists using averaged methods. Existence of exact solution in 3 dimension is what some people are working on. Then a question may arise that do quantum fluids obey the Navier strokes equation. The answer is yes. For quantum fluids, there exists a set of two equations called Madelung's Equations 2 , which is a quantum variant of Navier Strokes equation and continuity equation for classical fluids. Madelung's Equations are semi-classical equations that tell about the dynamics of Quantum Fluids. But "is this enough?", or we need a full quantum equation for defining the dynamics of quantum fluids.
Examples of Quantum Fluids include the presence of a large number of neutrons in a comparatively small volume in neutron stars. They deviate from Maxwell-Boltzmann statistics and follow Fermi-Dirac Statistics, which is one part of Quantum Statistical Mechanics. Another common example is Bose-Einstein's Condensates that do not follow Maxwell Boltzmann Statistics. To explain these large variety of fluids that are important in condensed matter physics and astrophysics, requires a new equation that will explain the dynamics of these systems. Some important concepts used to work on this project involves ideas of mutliparticle eigenfunction because this was obviously unavoidable. Fluids are treated as continum media and hence it is convenient to talk about field. In classical case, it is velocity field which is a type of vector field. The new equation starts with describing multiparticle wavefunction and then observing how this idea of discrete multiparticle can be systematically mapped to cumulative behaviour.

A. Describing the eigen-function of Quantum Fluids
Taking a brief look at II will tell that the number density of any Quantum Fluids can be analogously related to the mod square of the quantum mechanical wavefunction. Hence the normalization constant that appears in Schrodinger equation 3 is no more a constant but square root of number density which is a function of spatial coordinates and time. This idea can be fetched on philosophical grounds of Copenhagen interpretation. Born Rule says that for a single particle . P(V) is probability of finding a particle within volume V. Suppose the whole system is actually confined in the given volume, then P is unity. Therefore, The above formulation will help us on building the work further. It is mentioned in abstract section that, to solve the dynamics of Quantum fluid we need two fields, one is velocity field and other one is density field. So it is important to connect the velocity field with some analogous variable in the Schrodinger equation. It appears in the calculation in II that the phase of the quantum mechanical wavefunction is actually related to the velocity field. Therefore, the analogous parameter to eigenfunction is Schrondinger Equation for Quantum fluids can be written as , where x µ denotes the spatial coordinates for µ = 1, 2, 3. And κ is a constant.
Where ρ m is number density and it is interpreted as |ψ| 2 . The detailed analogy of this is given in subsection I A and the mathematical description is in II. This approach is clearly a semiclassical approach as Schrodinger equation is not applied rather the interpretation of Schrodinger equation is taken into account

II. MATH AND EQUATIONS
The very famous time dependent Schrodinger equation, that defines non relativistic Quantum mechanics for many body system is written as Where Ψ is multiparticle wave function defined as product of individual wavefunctions. Hydrodynamic Equation of Continuity is written as The complex conjugate of ψ is ψ †. So, Equation (3) If we multiply Equation (3) with ψ † and Equation (5) with ψ and subtracting Equation (5) from Equation (3) ih See that equation (7) is rearrangement of Equation (6), such that Equation (6) (1)). It must be noted that in continuity equation only one partial differential with respect to spatial co-ordinate(s) appear(s), but here there is a sum of partial differentials with respect to the coordinates of individual particles. This can be solved if we take a small infinitesimal volume element with is composed of N particles and the differentials with respect to individual coordinates can be transformed to centre of mass coordinate. So, so the sum of all the partial derivates equal partial derivate with respect to COM coordinate. Therefore, |ψ| 2 = ρ m . Now the interesting part of comparing Equation (4) with Equation (7), is the comparision of the second term in both equations.
The wave function in Quantum mechanics is usually written as ψ = Aexp(ikx − iωt). Therefore, Equation (8) changes to Writing u.dr ≈ u.r. Therefore wavefunction in Quantum Mechanics can to fluid dynamics variable is

A. Schrodinger-Navier Strokes Equation for Quantum Fluids
Refer to Equation (3). partial derivative with respect to time must Change to material derivative or Laplace Derivative such that So the new form of Equation (3) contains material dertivate instead of partial derivate.
The potential V contain the potential from Navier Stroke's Equation as well as Bohm's potential. Quantum fluids are believed to be inviscid fluids. Therefore, their kinectic viscosity is zero. Putting all this conditions Navier Stroke's equation looks like The left side of Equation (13) is the force per unit volume due to kinetic Energy and the right side denotes the for force per unit volume due to potential energy term. I will simply call therm kinetic and potential force for simplicity. Force can be written as negative gradiant of of potential energy.Now, V denote Potential Energy and V is for Volume. This just chosen to remove the confusion due to notation. Right hand sides of Equation must be equal.
Put Equation (15) on Equation (12) ih (16) I can put the final expression, replacing ψ in equation (16) with Equation (10). This will be a very simple but tedious calculation. Therefore, it is useful to write the final expression directly.
The final equation can be broken into two part: one that contain iota as a factor and second is the real part. Equation (16) does not include the spin as in the absence of magnetic field spin remains degenerate. But system with local or global magnetic field like neutron stars, Pauli's Equation 4 has to be used. This implies that quantum fluids have intrinsic spin, which has no classical analog. This article does not go that further but the point remains interesting and opens a new phenomenon for fluids. What spin of a fluid mean probably has no explanation, but it does mean that some different and strange behaviour can be observed in the case of Quantum Fluids. For spin 1/2 fluids there must be two different densities.
A pure Quantum equation has been proposed taking madelung's equation into consideration along with Schrodinger Equation. The equation is proposed to govern the dynamics of quantum fluids. this has to be done by correlating Wavefunction continuity equation and Continuity equation. Calculations showed that in case of quantum fluids the wavefunction can be replaced by ρ m exp 4iπ 2 m h u.r . Bohm's potential automatically, appear in the equation, therefore it has not been used externally, only Navier Strokes potential terms have been added.