## Abstract

A weakly non-local extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit, the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of the Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.

## 1. Introduction

Weakly non-local, coarse-grained, phase field and gradient are attributes of theories from different fields of physics, indicating that in contradistinction the traditional treatments, the governing equations of the theory depend on higher-order gradients of the state variables. Weakly non-local is a nomination in continuum physics dealing with internal structures (Mariano 2002), coarse-grained or phase field appears in statistically motivated thermodynamics (e.g. Hohenberg & Halperin 1977; Penrose & Fife 1990) and gradient is frequently used in mechanics (e.g. Cimmelli & Kosiński 1997; Béda 2000). The simplest way to find weakly non-local equations can be exemplified by the Ginzburg–Landau equation, which can be considered as a first weakly non-local extension of a homogeneous relaxation equation of an internal variable. The traditional derivation of the Ginzburg–Landau equation is based on a characteristic mixing of variational and thermodynamic considerations. One applies a variational principle for the static part, and the functional derivatives are introduced as thermodynamic forces into a relaxation type equation. A clear variational derivation to obtain a first-order differential equation is impossible without any further ado (e.g. without introducing new variables to avoid the first-order time derivative, which is not a symmetric operator; e.g. Ván & Muschik 1995). One can apply these kinds of arguments in continuum theories, in general, preserving the doubled theoretical framework separating reversible and irreversible parts of the equations (e.g. Grmela & Öttinger 1997; Öttinger & Grmela 1997). However, there are also other attempts to unify the two parts with different additional hypotheses (e.g. Gurtin 1996; Mariano 2002) and to eliminate this inconsistency of the traditional approach.

The ultimate aim is to find a unified and predictive theoretical framework that involves higher-order gradients in the governing equations of physics. Most of the approaches mentioned above are systematic in the sense that the gradient-dependent terms are not introduced in an *ad hoc* way and the Second Law of Thermodynamics plays an important role. An analysis of the involved thermodynamics given by Ván (2005*a*,*b*) shows that in the case of the Ginzburg–Landau equation, one can determine the variational part purely from the requirement of a non-negative entropy production, without any additional assumptions. The derivation based on the Second Law shows why, in what sense and under what conditions, the Ginzburg–Landau form is distinguished among the possible other weakly non-local equations of a single internal variable.

However, the Ginzburg–Landau equation is only a prototypical example and several different weakly non-local differential equations can appear in continuum physics. In this paper, we investigate quantum mechanics from this point of view. It is well known that classical quantum mechanics is a non-local field theory. Moreover, non-local in two different senses. One kind of non-locality is embodied in the family of EPR (Einstein–Podolsky–Rosen) experiments, expressing the appearance of conservation laws in a probabilistic theory. At the same time, quantum mechanics is a weakly non-local theory because the Bohmian quantum potential depends on the derivatives of the quantum probability density (e.g. Bohm 1951; de la Peňa & Cetto 1996). The hydrodynamic form of quantum mechanics is a special weakly non-local fluid mechanics.

In this paper, we give all possible weakly non-local extensions of fluid mechanics that are compatible with the Second Law. Introducing the basic and constitutive state spaces below, we have developed a theory, which is in agreement with the Second Law, disregarding the problem of material frame independence. Although the basic balances at the beginning and the final constitutive function of reversible pressure are frame independent, the whole procedure is based on frame-dependent quantities. One cannot neglect this difficult and important problem, which is to be treated in a forthcoming paper.

A straightforward application of Liu's procedure gives the surprising consequence that a weakly non-local extension of traditional fluid mechanics in the density leads to a generalization of the Euler equation that incorporates the hydrodynamic model of quantum mechanics. Moreover, we will show that this model is distinguished among the different possible weakly non-local fluids in the sense that the characteristic Fisher information like form of the gradient-dependent part of the entropy density function is a unique consequence of its isotropic and additivity properties. After these considerations, we treat briefly some interpretational and conceptual problems, too. We argue that our investigations lead to a derivation of quantum mechanics that is based on a minimal number of assumptions.

The starting point of our investigations is fluid mechanics in a general sense. However, we do not consider fluid mechanics as a phenomenological theory with a rich and definite microscopic-molecular background, but rather as an empty bottle of general physical principles that are valid in any field theory (e.g. balance of momentum, conservation of mass, Second Law, etc.). For example, the balances are assumed to be valid on a microscopic scale as in quantum field theories. The approach of non-equilibrium thermodynamics fills this bottle with content using a step-by-step procedure. First one considers only the general principles and exploits them, and after that introduces the specific, material-dependent assumptions. Therefore, arriving at the so-called hydrodynamic model of quantum mechanics does not prevent us from transforming it to a wave function form and enjoying the advantages of a linear equation. However, we should know the physical content of the linearity and should be aware of the price. For example, it is well known that the Schrödinger equation is not a reference frame independent, objective equation. Moreover, it cannot be written in a frame-independent form (e.g. Matolcsi 1984, 1986) because of its energetic origin. Similarly, canonical quantization is also frame/observer dependent (e.g. Bolivar 1998). However, the theories based on a momentum balance (e.g. the hydrodynamic model) are frame independent and objective, as shown by Fülöp & Katz (1998).

## 2. Fluid mechanics in general

The *basic state space* of one-component fluid mechanics is spanned by the density *ρ* and the velocity * v* of the fluid. Hydrodynamics is based on the balance of mass and the balance of momentum (e.g. Gyarmati 1970). Classical fluid mechanics is the theory, where the constitutive space, the domain of the constitutive functions, is spanned by the basic state space (

*ρ*,

*) and the gradient of the velocity ∇*

**v***. The pressure/stress tensor is the only constitutive quantity in the theory. The pressure function defines the material in continuum physics. For example, it determines whether it is a fluid or a solid, and whether it is a material in local equilibrium without memory or not, etc. Introducing higher-order gradients of the basic variables into the constitutive space, one can obtain weakly non-local extensions regarding the density and the velocity. In this paper, we investigate the weakly non-local extension of classical fluid dynamics in the density.*

**v**The balance of mass in local substantial form can be written as(2.1)where *ρ* is the density, * v* is the velocity,

*σ*

_{m}is mass production, and the dot above the quantities denotes the substantial (material) time derivative.

The balance of momentum, i.e. the Cauchy equation, is(2.2)where * P* is the pressure and

*is the force density. For discussing the characteristics of continuum materials, the effects given by the source terms do not play any role. The Second Law requires that the production of the entropy is non-negative in insulated and source-free systems. Particularly in our case, if there is no production of mass (*

**f***σ*

_{m}=0) and there are no external forces (

*ρ*=

**f****0**), then the production of entropy must be non-negative(2.3)

The constitutive quantities in the above equations are the pressure * P*, the specific entropy

*s*and the conductive current

**j**_{s}of the entropy. According to the Coleman-Mizel form of the Second Law, the entropy inequality is a constitutive requirement, as analysed by Muschik & Ehrentraut (1996). That is, we are looking for constitutive functions that solve the inequality. Thus, the entropy inequality should be a pure material property; it is required to be valid independently of the initial conditions. In the exploitation of the inequality, Liu's theorem plays an important technical role. Details of the different state spaces, more detailed description of thermodynamic concepts and the applied mathematical methods (especially the Liu procedure) are given by, for example, Mushik

*et al*. (2001) and Ván (2003, 2005

*a*), who analysed the special requirements of a weakly non-local extension.

Let us remark that, in fluid systems, all balances of different (kinetic, potential, internal and total) energies should be considered (e.g. Verhás 1997). However, in our treatment, dealing with non-local extensions only in the density, temperature and internal energy changes do not play any role. Moreover, as we want to generalize the treatment of fluid mechanics, we avoided the computational complications coming from additional governing equations and formulated the Second Law with the single inequality (2.1) and (2.2)–(2.3). In the case of moving fluids, it is better to regard the function *s* as a generalized (kinetic) potential of Glansdorff & Prigogine (1971). However, we intentionally did not specify a complete classical thermodynamic background, where the meaning of the different physical quantities is fixed. Avoiding the unnecessary extension of the terminology and emphasizing the very thermodynamic point of view of the argumentation, we call the potential function entropy.

## 3. Non-local fluid mechanics: the Schrödinger–Madelung equation

Fluid mechanics is treated in the case when the *basic state space* is that of classical hydrodynamics (*ρ*, * v*) and the

*constitutive space*contains gradients of the density

*ρ*in addition to the classical case and is spanned by the variables (

*ρ*, ∇

*ρ*,

*, ∇*

**v***, ∇*

**v**^{2}

*ρ*). Here ∇

^{2}

*ρ*denotes the second space derivative of

*ρ*(sometimes written as ∇∘∇

*ρ*, where ∘ is the traditional notation of the tensorial/dyadic product in hydrodynamics). The

*space of independent variables*is spanned by the next time and space derivatives of the constitutive variables, that is, by (, , , , , ∇

^{2}

*, ∇*

**v**^{3}

*ρ*), as a consequence of the entropy inequality. It is easy to see that these quantities are independent, because all substantial derivatives contain different partial time derivatives and gradients of partial time derivatives in an additive manner, e.g. .

With introducing the above constitutive state spaces, we have defined non-local fluids, which are well known in fluid mechanics. Second grade fluids are weakly non-local both in the velocity and in the density. Weakly non-local fluids in the density are called Korteweg fluids (Korteweg 1901). A thermodynamic theory of Korteweg fluids was developed by Dunn & Serrin (1985). In our investigations, the entropy current plays a similar role to *interstitial working* in their theory. There are several examples of the applicability of generalized fluid models (e.g. Galdi 2000).

In applying Liu's procedure, the constraints are the balance of mass (2.1) and the balance of momentum (2.2). Moreover, because of the higher derivatives of density in the constitutive space, the space derivative of the mass balance is also to be considered as a constraint,(3.1)This situation is similar to what happened in the case of the thermodynamic derivations of the Ginzburg–Landau equation (Ván 2005*b*), or in relativistic constitutive theories (Herrmann *et al*. 1998). On the other hand, higher-order space derivatives of equation (3.1) cannot be constraints, because they contain derivatives that have already been included in the space of independent variables.

Now we apply Liu procedure (Liu 1972), with the method of Lagrange–Farkas multipliers asHere the multipliers *Γ*_{1}, *Γ*_{2} and *Γ*_{3} were introduced for the constraints (2.1), (2.2) and (3.1), respectively. The subscript numbers denote derivation according to the corresponding variable in the constitutive space (*ρ*, ∇*ρ*, * v*, ∇

*, ∇*

**v**^{2}

*ρ*), e.g.

*D*

_{1}

*f*=∂

*f*/∂

*ρ*. The source terms in the balances have been considered as zero. For what follows, it is important to observe that the substantial time derivative does not commute with the space derivative (gradient), instead, the following identities are to be applied:For the sake of easier applicability we give these equations with indices, too:Here

*i*,

*j*,

*k*=1, 2, 3 denote the Cartesian coordinates. Using these identities the terms in the above inequality can be rearranged as

The multipliers of the independent variables are the Liu equations, respectively(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)Here, the last two equations are given with indices to avoid misunderstanding. Equation (3.8) is symmetric in every tensorial component (for all permutations of *j*, *k* and *l*), and (3.7) is symmetric in *j* and *k* because of the symmetry of the corresponding independent variables. In the following, we want to solve Liu's equations (3.2)–(3.7). As a consequence of (3.5) and (3.6), the specific entropy does not depend on the second gradient of *ρ* and on the gradient of * v*. Hence,

*s*(

*ρ*, ∇

*ρ*,

*, ∇*

**v***, ∇*

**v**^{2}

*ρ*)=

*s*(ρ, ∇

*ρ*,

*). Equations (3.2)–(3.4) give the Lagrange–Farkas multipliers as derivatives of the entropy. Therefore, from a thermodynamic point of view, they are a kind of generalized intensive variables in the theory, as pointed out by Kirchner & Hutter (2002). Let us look at the entropy as the primary physical quantity, i.e. we want to express the other constitutive functions with its help. Now, one can give a solution of (3.8) and (3.7) aswhere*

**v**

**j**_{0}is an arbitrary (differentiable) function. Thus, Liu's equations can be solved and yield the Lagrange–Farkas multipliers as well as restrictions for the entropy and the entropy current. Applying these solutions of the Liu equations, the dissipation inequality can be written asHere denotes the second-order unit tensor (

*δ*

_{ij}). Let us now define a

*traditional fluid*with a specific entropy of the following form:(3.9)From this form, it is clear that

*s*

_{s}corresponds to a static (equilibrium) specific entropy and

*s*is a kind of general non-equilibrium potential closely connected to the kinetic potential of Glansdorff & Prigogine (1971). However, here we exploited the entropic representation of the variables and

*s*

_{s}also depends on the gradient of the density function. Now, it is reasonable to introduce a new notation of the derivatives as

*∂*

_{ρ}≔

*D*

_{1}and

*∂*

_{∇ρ}≔

*D*

_{2}.

If **j**_{0}=**0**, as usual, then we can write the dissipation inequality and the entropy current as(3.10)

(3.11)

The advantage of the entropy (3.9) of a traditional fluid is that the inequality (3.10) is solvable, because it has the force–current form of irreversible thermodynamics and contains the pressure as a single dynamic constitutive function (*s*_{s} is static and assumed to be known). We can define the *non-local reversible pressure* as(3.12)If the total pressure is of this form, then the entropy production is zero; there is no dissipation, the theory is reversible (conservative). If the entropy is local (independent of the gradient of the density), then we obtain(3.13)therefore, the corresponding equations are of the ideal Euler fluid, where *p*(*ρ*)=*ρ*^{2}*∂*_{ρ}*s*_{s}(*ρ*) is the scalar pressure function. From this form, we can identify the relation of our generalized potential and the traditional entropy.

Introducing the viscous pressure **P**^{v} as usual, we can solve the dissipation inequality (3.10) and give the corresponding Onsagerian conductivity equation asHere * L* is a non-negative constitutive function. Let us recognize that if

*s*

_{s}is independent of the gradient of the density,

*is constant, and*

**L**

**P**^{v}is an isotropic function of only ∇

*, then we obtain the traditional Navier–Stokes fluid.*

**v**One can prove easily that the reversible part of the pressure of a traditional fluid is *potentializable*, i.e. there is a scalar valued function *U* such that(3.14)*U* can be calculated from the entropy function as(3.15)Therefore, in case of reversible fluids, the momentum balance can be written alternatively as(3.16)

Giving the form of *s*_{s}, we obtain some specific non-local fluids.

*Schrödinger–Madelung fluid*. Here the entropy is defined as(3.17)where *ν*_{SchM} is a constant scalar. The corresponding reversible pressure is(3.18)where ∘ denotes the tensorial/dyadic product, as mentioned before. The potential is(3.19)where we introduced to show more clearly that (3.19) is the quantum potential in the de Broglie–Bohm version of quantum mechanics (if ; see Bohm 1951; Holland 1993).

Further, the entropy current of the Schrödinger–Madelung fluid is(3.20)

In the hydrodynamic model of quantum mechanics, the pressure cannot be determined uniquely, because one concludes from the Schrödinger equation that a neutral quantum fluid preserves the vorticity. Therefore, in the hydrodynamic model, the pressure is calculated from the potentializability condition (3.14) (from the potential) and, thus, one can add a curl of any function of the density and its gradient without changing the physics (Bernoulli equation). For example, replacing Δ*ρ*+∇^{2}*ρ* with 2∇^{2}*ρ* in (3.18), one obtains the Jánossy–Ziegler pressure (Jánossy & Ziegler 1963; Harvey 1966). Holland (2003) recognized that it is a (gauge) freedom in case of neutral fluids, but it can be important when vorticity is not zero. In our thermodynamic derivation, the pressure is the primary constitutive quantity, it is determined uniquely and the potentializability is the consequence of the thermodynamic structure.

In the following, we introduce two other fluids of the Korteweg family. They serve as illustrative, mathematical examples to understand the most important specific property of Schrödinger–Madelung fluids, the existence of quantized solutions. The Landau and the alternative fluids show that these kind of solutions can also appear in other fluids, demonstrating that the existence of quantized solutions is not a distinctive property of the Schrödinger–Madelung fluid.

*Landau fluid*. The simplest globally concave and isotropic entropy depends on the square of the density gradient. Owing to the similarity to the Ginzburg–Landau free energy density, we will call Ginzburg–Landau fluid the material that is defined by the following form of the entropy(3.21)where *ν*_{Lan} is a constant coefficient. The corresponding reversible pressure function is(3.22)

The potential is(3.23)

The entropy current can be also given as(3.24)

*Alternative fluid*. The potential has the simplest form, if the entropy is written as(3.25)where *ν*_{Alt} is a constant coefficient. In this case, the reversible pressure is(3.26)and the non-local term in the potential is simply(3.27)Finally, the entropy current can be written as(3.28)

## 4. The origin of quantum potential

An important property of the Schrödinger–Madelung fluid is that if the motion of the fluid is vorticity free, ∇×* v*=

**0**, then the mass and momentum balances can be transformed into and united in the Schrödinger equation. Hence, the balance of momentum (2.2) can be derived from a Bernoulli equation (in a given inertial reference frame). Defining a scalar-valued phase (velocity potential) bythe Bernoulli equation is obtained from the second part of (3.16) written as(4.1)

Then, introducing a single complex-valued function ψ≔*R* e^{iS} that unifies and *S*, it is easy to find that sum of (2.1) multiplied by and (4.1) multiplied by *mR* e^{iS} form together the Schrödinger equation for free particles(4.2)Therefore, what we have accomplished is a kind of derivation of the quantum mechanics in a very general framework. Moreover, we have obtained some more different, but related, fluid models. At this point, several questions can arise. What is the relation of this thermodynamic derivation to other approaches? Are there ’quantized’ solutions of the Landau or alternative fluids under similar conditions to that in quantum mechanics? Are there any distinctive properties of the Schrödinger–Madelung model among the other weakly non-local fluids?

First of all, to understand the results of Liu procedure from a different point of view, let us observe that equation (3.15) has an Euler–Lagrange form. The corresponding Lagrangian density is . Moreover, there are attempts to get mass and momentum balances from a variational principle in case of rotational free motion (e.g. Spiegel 1980; Reginatto 1998) and rotation (Wagner 2005). In our thermodynamic train of thought, a variational principle and an Euler–Lagrange form was not a starting point, but a consequence of the Second Law in a special reversible case.

It is interesting to observe that the Schrödinger–Madelung model is not distinguished from the other two fluids in producing static ‘quantum’ solutions. The static fluid dynamic equations give a generalized eigenvalue problem(4.3)

Because *U* is given as a functional derivative of *ρs*_{s}, the existence of multiple solutions is connected to the concavity properties of the entropy function. Among the previously defined three fluids, only the Landau fluid preserves the global concavity of the entropy both in the density and in the gradient of the density. One can calculate easily that the second derivatives of the entropies of the Schrödinger–Madelung fluid and of the alternative fluid are positive semi-definite. The Schrödinger–Madelung fluid has a rich structure of stationary solutions in the case of simple conservative force fields (the same as for the Schrödinger equation). According to Ván & Fülöp (2004), this property seems to be connected to the semi-definite property of the entropy, the existence of similar ‘quantized solutions’ can be expected in the alternative fluid, too. This is an interesting question both from a physical and from a mathematical point of view. Evidently, a large set of different generalized eigenvalue problems emerge.

Let us remark here that the concavity of the entropy (kinetic potential) is also connected to the stability properties of the equilibrium solutions of the fluid dynamic equations (e.g. Ván & Fülöp 2004).

Finally, we turn our attention to the form of the static part of the specific entropy function. Apart from a constant multiplier, it can be written as , which is the trace of the Fisher information tensor of the probability density *ρ* (e.g. Borovkov 1998). Here we will show shortly that this special form is a straightforward consequence of some general properties of the theory and of the entropy function *s*_{s} (*ρ*, ∇*ρ*). In the following arguments, we require some basic properties of point masses, encoded in a continuum theory.

First of all, the entropy is *isotropic*. An isotropic function *s*_{s} of *ρ* and ∇*ρ* has the following property (e.g. Pipkin & Rivlin 1959; Smith 1964)(4.4)On the other hand, for independent particles, one requires the separability of the governing equations, therefore, the *additivity* of the entropy function in a definite way. We postulate that, in case of independent particles, the governing equation should be additively separable and, at the same time, the probability interpretation has to be preserved when considering the entropy of the system of particles. This is exactly the physical property, and why one requires the linearity of the Schrödinger equation and any of its dissipative extensions written in probability amplitude (wave function) variables.

For the sake of simplicity, we restrict ourselves to two particles. The generalization to finite number of independent particles is straightforward. For two distinct particles we can consider the two-particle probability density *ρ*(**x**_{1}, **x**_{2}), which is the product of the one-particle probability densities *ρ*_{1}(**x**_{1}) and *ρ*_{2}(**x**_{2}). Then the gradient of *ρ* is meant in two variables, too. Thus, we have and—omitting the variables **x**_{1} and **x**_{2}—. As a consequence, in case of isotropic entropies, the additivity requirement can be written as(4.5)

If is continuously differentiable, then differentiating the above equality by (∇*ρ*_{1})^{2} and (∇*ρ*_{2})^{2}, respectively, we have that

Here *D*_{2} denotes the partial derivative of by its second argument. Therefore,from which it follows that(4.6)where is an arbitrary function (the local part of the entropy). Repeating the above argument with the derivatives by the first argument of , one finds thatConsequently, , where *s*_{0}=0, according to the additivity. Therefore,(4.7)The first term has the form of a Fisher information and the second term has the form of a Shannon information measure. The solution is unique with the above requirements.

Let us now consider that our probability continuum should be a theory of particles. Therefore, we require the *mass-scale invariance* of the entropy, i.e. the entropy density is a first-order homogeneous function of the probability density. In this case, the specific entropy is scale invariant, a zeroth-order homogeneous function of the density(4.8)for any real number *λ*. This is a necessary condition to have a particle interpretation of a continuum. In this case, the quantum potential is independent of the mass-scale and, thus, the equations of free motion also have the same property. One can see this clearly from the particle equation form of the momentum balance (the second part of 3.16). Considering this condition the mass, the measure of the inertia of the continuum acts uniformly, independently of the space coordinates. With using specific entropy that depends only on the specific quantities (the density is the reciprocal value of the specific volume), we required a volume-scale invariance of the corresponding functions. The mass-scale invariance is something similar; it is a kind of mass-extensivity.

Mass-scale invariance requires that *k*=0 in (4.7), excluding the logarithmic part.

It is interesting to note that there are researches regarding a kind of weakly non-local statistical physics. It investigates the foundations of equilibrium probability distributions and static non-local extensions of fundamental equations of mathematical physics from the point of view of Fisher information instead of the traditional Boltzmann–Gibbs–Shannon one (see Frieden 1990; Plastino & Plastino 1995; Frieden *et al*. 1999, 2002). However, the relationship between Fisher information and thermodynamic entropy in non-equilibrium situations is not clear and, especially, the non-negativity of the entropy production is questioned by Nettleton (2002, 2003). In the light of the recent investigations, for the case of the Schrödinger equation, the entropy inequality is fulfilled as an equality.

## 5. Conclusions

In this paper, we have shown that the Second Law of thermodynamics restricts considerably the possible pressure functions of fluids that are weakly non-local in density. Several different traditional fluids were defined, where the non-equilibrium specific entropy is additively quadratic in the velocity (3.9), by some simple possible forms of the non-local part of the entropy function. In the conservative limit, when the entropy production is zero, we have found that the entropy density is a Lagrangian of the gradient-dependent potentials; therefore, in the case of vorticity free motion, it can be substituted into a Seliger–Whitham-type variational principle (see Seliger & Whitham 1968; Spiegel 1980). However, in our thermodynamic approach, the Euler–Lagrange form was a consequence of the Second Law in the non-dissipative limit. There was no need to refer to any variational principle at all.

There are two arguments for using the Second Law in microscopic considerations. First of all, one should consider that there are deeper levels in the microworld below quantum mechanics. There are always deeper levels. Therefore, it is not without merit to exploit the *existence* of these levels with a phenomenological argumentation, without referring to their particular structure (heat bath, vacuum fluctuations, etc.). On the other hand, in the above reasoning, the Second Law also restricted the structure of the equations of motion in the *reversible* limit.

We have seen that the Schrödinger–Madelung fluid, the hydrodynamic model of quantum mechanics, plays a distinguished role among the thermodynamically possible weakly non-local fluids. It is the only model where the non-local part of the entropy function is isotropic, additive and mass-scale invariant. All these three properties have a clear physical meaning. We need an additive entropy to obtain independent equations for independent free particles. Additivity requires isotropy. We need a mass-scale invariant entropy to have a mass that measures the inertia of the particle in our continuum theory. Only in this case will the inertia of the generalized ‘fluid’ be uniform and proportional to the force.

From a fluid mechanical point of view, the Schrödinger equation appears as a complex formulation of the coupled Bernoulli equation and the mass balance and has the great advantage of being linear. On the other hand, we should emphasize that our derivation of the hydrodynamic model of quantum mechanics was completely independent of the Schrödinger equation, and quantum mechanics appeared only as a special fluid with remarkable properties. Generalized Schrödinger-type equations appear as structure forming equations in several fields without any connection to quantum mechanics (e.g. Aranson & Kramer 2002). The possibility to transform generalized weakly non-local fluids into a well-known linear complex form is an important advantage in several investigations (as, for example, in cosmology, e.g. Widrow & Kaiser 1993; Coles 2002; Szapudi & Kaiser 2003).

Let us remark here that the usual homogenization procedures of classical field theories sometimes are used to eliminate gradient terms, to smooth the microstructural contributions; therefore, in our case, could replace the Ehrenfest theorem of quantum mechanics.

Let us make some remarks on other derivations of the basic equation of quantum mechanics. As it is well known, Schrödinger himself did not derive the equation. His suggestion is based on analogies and is justified by its consequences. The de Broglie–Bohm form or the hydrodynamic model of Madelung all start from the Schrödinger equation, so they are not derivations only, but different points of view given for interpretational purposes (nevertheless important and thought-provoking ones).

The stochastic model (e.g. Fényes 1952; Nelson 1966; de la Peňa & Cetto 1996; Fritsche & Haugk 2003; Garbaczewski 2003) provides a kind of derivation. Here one assumes a background random field, the active role of ‘vacuum fluctuations’ as an origin of the quantum potential, but one should not forget about the special assumed properties. The introduced stochastic velocity has a particular form, being proportional to the gradient of *ρ*, which form is motivated by the analogy (!) with diffusion. Here non-locality is disguised as a kind of velocity, hence requires a special structure and is similar to—results in—the mass-scale invariance of the entropy.

Non-local kinetic theories (e.g. that of Kaniadakis (2002)) all assume a definite microscopic background.

In §4, we have mentioned other approaches based on the Fisher information measure. These start from a variational principle and postulate the form of quantum term (as minimum Fisher information; e.g. Reginatto 1998) and interpret it with information theoretical ideas based on the measurement process (Frieden 1998). Such an approach is, thus, less a derivation, but rather a different interpretation (nevertheless important and thought-provoking).

A remarkable exception is the approach of Hall & Reginatto (2002*a*,*b*), who give arguments on the form of the entropy (in their formulation it is an additional term in the Lagrangian due to fluctuations) with a reasoning that is similar to ours. They require additivity and isotropy (implicitly), but instead of mass-scale invariance they use a different principle what they call ‘exact uncertainty’. With these requirements, they arrive at the same Fisher form of the quantum part of the entropy function. On the other hand, they start from a variational approach, which is a consequence of the Second Law in our considerations.

We have to emphasize again that our derivation here is based on general principles (Second Law) and requirements (additivity) and, therefore, is independent on any kind of interpretational issues of quantum mechanics. It contains a minimal set of assumptions that one can gain on a phenomenological level about one component fluid systems. It is surprising that the only thing in the equations that one should determine from microscopic considerations (from experiments) is the value of the Planck constant.

Our approach is similar that of Jaynes (1957) to equilibrium statistical physics both on the dynamic and on the static level. On the dynamic level, we used the entropy inequality—a part of the Second Law—and the basic balances as constraints to extract the maximum amount of information regarding the structure of a definite physical system. We have derived serious restrictions on the constitutive functions. Moreover, the reasoning is predictive in the Jaynesian sense. For example, in the dynamic case requiring non-negative entropy production, one can give definite predictions on the structure of non-local equations of multicomponent fluids, too. Considering phase separation as a constraint, Ván (2004*a*,*b*) constructed promising new models for granular and porous media.

Moreover, regarding the static part, our reasoning is completely analogous to the Jaynesian, phenomenological approach of statistical physics. Jaynes's arguments are based on the uniqueness of Shannon's information measure with some expected properties that are common with the required properties of the entropy function (extensivity, additivity). However, Shannon's proof was related to a discrete probability space and exploited the fact that the entropy is a composite (local) function. Here, we can regard our approach as a kind of non-local extension of the Jaynes–Shannon argumentation. The form (4.7) of the entropy is unique with the given requirements, therefore one can use it as a starting point. The emergent structure is independent of the microscopic background. Every reasonable microscopic approach (kinetic or stochastic) corresponds to the formulated general requirements.

Moreover, in (4.7) one can recognize the central formula of the extreme physical information principle of Frieden (1998), and our argumentation gives a kind of foundation of the principle (uniqueness) and also some limitations (conditions of validity as mass-scale invariance) and a different interpretation (there is no need of the measurement–information arguments, objectivity of the theory is reconstructed). Here the predictivity of the approach is even more evident as one can see from the increasing number of applications of statistical physics based on Fisher information.

## Acknowledgments

This research was supported by OTKA T034715, T048489 and T034603 and by Vásárhelyi and Partner Ltd. The authors wish to say thanks to Tamás Matolcsi, from whom they have learned that quantum mechanics is a closed but unfinished theory, and thank J. Verhás, S. Katz and K. Oláh for the discussions regarding several different topics of physics, including the foundations of quantum mechanics and thermodynamics. Thanks to the referees for their remarks and suggestions, especially for calling our attention to Korteweg fluids.

## Footnotes

- Received July 4, 2005.
- Accepted October 10, 2005.

- © 2005 The Royal Society