On the dynamical nature of nonlinear coupling of logarithmic quantum wave equation, Everett-Hirschman entropy and temperature

TL;DR

This paper explores the nonlinear coupling in a logarithmic quantum wave equation, linking it to thermodynamic temperature and quantum information entropy, and proposes a model connecting quantum mechanics, field theory, and thermodynamics.

Contribution

It introduces a combined quantum-mechanical and field-theoretical model with variable nonlinear coupling related to temperature and entropy, offering new insights into quantum thermodynamics.

Findings

The nonlinear coupling relates to temperature and entropy in quantum systems.

The model describes both nonlinear and linear quantum systems with external potentials.

Connections to Landauer's principle are discussed.

Abstract

We study the dynamical behavior of nonlinear coupling in a quantum wave equation of a logarithmic type. Using statistical mechanical arguments for a large class of many-body systems, this coupling is shown to be related to temperature which is a thermodynamic conjugate to the Everett-Hirschman's quantum information entropy. A combined quantum-mechanical and field-theoretical model is proposed, which leads to a logarithmic equation with variable nonlinear coupling. We study its properties and present arguments regarding its nature and interpretation, including the connection to Landauer's principle. We also demonstrate that our model is able to describe linear quantum-mechanical systems with shape-changing external potentials.

Equations

$$i\hslash {\partial }_t\Psi$$

$$b=b(\vec{r},t)$$

$$⟨\Psi |\Psi ⟩\equiv \underset{\mathcal{V}}{\int }|\Psi {|}^2{d}^3\vec{r}=N⩾1,$$

$$b(\vec{r},t)=b_0=\text{const},$$

$$S_{\Psi }=-⟨\Psi |\ln (a^3|\Psi {|}^2)|\Psi ⟩=-\underset{\mathcal{V}}{\int }|\Psi {|}^2\ln (a^3|\Psi {|}^2)d^3\vec{r},$$

$${\cal U}_{\Psi}=\langle\Psi|\hat{{\bf H}}|\Psi\rangle+\Delta T_{\Psi}\,S_{\Psi},\ \$$

$$\Delta T_{\Psi }={\left(\frac{\partial {\mathcal{U}}_{\Psi }}{\partial S_{\Psi }}\right)}_{\mathcal{V}}\propto b,$$

$$I_{\Psi }=-{\log }_2(a^3|\Psi {|}^2)=-\ln (a^3|\Psi {|}^2)/\ln 2.$$

$$P\propto \exp (-\mathcal{E}/T)\approx \exp (-\mathrm{U}/T),$$

$$⟨x|\hat{\mathrm{U}}|\Psi ⟩=\hat{\mathrm{U}}\Psi (\vec{r},t)=-K(T-T_0)\ln (A|\Psi (\vec{r},t){|}^2)\Psi (\vec{r},t),$$

$$\hat{\mathrm{H}}|\Psi ⟩=\left(\frac{{\hat{\vec{p}}}^2}{2m}+\hat{\mathrm{U}}+V_{\text{ext}}\right)|\Psi ⟩,$$

$$b\sim T\sim T_{\Psi },$$

$$b\ln \left(a^3|\Psi {|}^2\right)\Psi =b\left(a^3|\Psi {|}^2-1\right)\Psi +\dots ,$$

$$b=b\left(\vec{n}\cdot \vec{\nabla }\sigma ,\vec{\nabla }\sigma \cdot \vec{\nabla }\sigma ,...\right),$$

$$i\hslash {\partial }_t\Psi$$

$${\nabla }^2\sigma$$

$$\kappa {\rho }_{\sigma }=f(\rho ),$$

$$\sigma =\sigma (r,t),$$

$$i\hslash {\partial }_t\Psi +\frac{{\hslash }^2}{2m}{\nabla }^2\Psi +{\partial }_r\sigma \ln (a^3|\Psi {|}^2)\Psi =0,$$

$${\nabla }^2\sigma ={\nabla }_r^2\sigma =4\pi f(|\Psi {|}^2),$$

$${\rho }_{\sigma }\propto f(|\Psi {|}^2)\sim \delta (r)+\sum _{n=1}^{\infty }\frac{a_n}{r^n},$$

$${\nabla }_r^2\sigma =-4\pi \left[Q_{\sigma }\delta (r)-\frac{b_0}{2\pi r}+\mathcal{O}(1/r^2)\right],$$

$$\sigma ={\sigma }_0+\frac{Q_{\sigma }}{r}+b_{0}r,$$

$$i\hslash {\partial }_t\Psi +\frac{{\hslash }^2}{2m}{\nabla }^2\Psi +\left(b_0-\frac{Q_{\sigma }}{r^2}\right)\ln (a^3|\Psi {|}^2)\Psi =0,$$

$$\vec{r}{}^{\prime }=\vec{r}/a,t^{\prime }=t/\tau ,\tilde{\Psi }=a^{3/2}\Psi ,$$

$$i{\partial }_{t^{\prime }}\tilde{\Psi }+{\nabla }^{\prime 2}\tilde{\Psi }+\left({\tilde{b}}_0-\frac{\tilde{q}}{r^{\prime 2}}\right)\ln (|\tilde{\Psi }{|}^2)\tilde{\Psi }=0,$$

$$\tilde{\Psi }(\vec{r},t)=\exp (-i\tilde{\omega }t)\psi (\vec{r}),$$

$${\nabla }^2\psi +\left({\tilde{b}}_0-\frac{\tilde{q}}{r^2}\right)\ln (|\psi {|}^2)\psi +\tilde{\omega }\psi =0,$$

$${\nabla }^2={\nabla }_r^2+\frac{1}{r^2}{\nabla }_{S^2}^2,$$

$${\tilde{V}}_{\text{eff}}(\vec{r})=\left(\frac{\tilde{q}}{r^2}-{\tilde{b}}_0\right)\ln (|{\psi }_s(\vec{r}){|}^2),$$

$${\tilde{s}}_{\Psi }^{\text{(r)}}=-\iint |\psi {|}^2\ln (|\psi {|}^2)r^2{\text{sin}}^2\theta d\theta d\varphi ,$$

Full text

††preprint: Z. Naturforsch. A 73, 619-628 (2018)  [DOI: 10.1515/zna-2018-0096]

I Introduction

Assuming three spatial dimensions (lower-dimensional cases can be studied by analogy), let us consider the logarithmic quantum wave equation, often referred also as the logarithmic Schrödinger equation (LogSE):

[TABLE]

where the coupling

[TABLE]

quantifies the strength of nonlinear self-interaction, $a$ is a constant parameter of dimensionality length required to make the argument of the logarithm dimensionless, $m$ is the particle’s mass, $V_{\text{ext}}=V_{\text{ext}}(\vec{r},t)$ is an external potential (sometimes dubbed the trap potential), and $\nabla^{2}=\vec{\nabla}\cdot\vec{\nabla}$ is the Laplacian (in the relativistic version of Eq. (1), the derivative part would be replaced by the d’Alembertian). The wavefunction $\Psi=\Psi(\vec{r},t)$ is assumed to be normalized to the number $N$:

[TABLE]

where ${\cal V}$ is the volume occupied by our system.

The simplest case of Eq. (2),

[TABLE]

was historically the first to be studied ros69 ; ros69a ; gg1 ; gg1a . The corresponding models were proven to be instrumental in dealing with extensions of quantum mechanics gg1 ; gg1a ; bbm79 , physics of quantum fields and particles ros69 ; ros69a ; em98 ; hkt10 ; Dzhunushaliev:2012zb ; gul14 ; gul15 ; dmz15 , optics and transport or diffusion phenomena gg2 ; gg6 , classical hydrodynamics of Korteweg-type materials ko1901 ; ds85 ; gg5 ; gg5a ; gl08 ; gl08a , nuclear physics gg3 ; gg4 , theory of dissipative systems and quantum information gg7 ; gg9 ; gg10 ; gg11 ; lo04 ; lm13 ; mw14 ; zrz17 , theory of quantum liquids and superfluidity az11 ; zlo12 ; bo15 ; btl16 ; z17zna , and theory of physical vacuum and classical and quantum gravity Zloshchastiev:2009zw ; gg15 ; gg14 ; szm16 . The mathematical properties of the logarithmic wave equation and its modifications and solutions were also extensively studied bbm79 ; az11 ; bo15 ; z17zna ; ch80 ; hh13 ; gs13 ; dms14 ; ss15 ; dsz15 ; js16 ; ard16 ; ard16a ; ard16b ; wct16 ; tz17 ; brz17 ; ns17 ; fon17 ; pg17 ; pa17 ; zrz17 ; sha17 ; as17 ; bcs18 , to mention just a few examples.

Notwithstanding the success of models with a constant coupling (4), there remain a few questions which need to be addressed.

First, what is the value of the coupling (2), is it a fundamental constant, or is it related to dynamical observables? Past studies gg1 ; gg1a ; gg3 ; gg4 ; gkz81 have shown that for a large class of conservative quantum systems the nonlinear term’s effect must be rather small. On the other hand, in the theory of superfluid He-4, which is an example of a system being in thermal contact with a reservoir of large heat capacity as to maintain constant temperature, this term plays a crucial role zlo12 . Therefore, it seems that this coupling can take different values, depending on prevailing physical conditions, i.e., it can vary from system to system. In other words, this means that its value cannot be a fundamental constant, but rather a dynamical function; while the formula (4) can be regarded as a first-order approximation or a limit value of this function.

Second, if this coupling is a nontrivial dynamical notion, then what is its physical meaning? One possible idea, which was advocated in Refs. gg9 ; az11 , is to relate it to a certain kind of temperature $T_{\Psi}$, which is a thermodynamical conjugate to a quantum analogue of Shannon information entropy sha48 ; sha48a , referred here as the Everett-Hirschman’s (EH) entropy ever55 ; hir57 . The latter being defined as

[TABLE]

where the Boltzmann constant is hereafter assumed to be $k_{B}=1$, and we adopt the sign conventions of Ref. sha48 ; sha48a . This entropy can be used as a measure of the localization of a system, or as an inverse measure of its extendedness, more details can be found in Ref. ever55 .

In this framework, the logarithmic quantum wave equation can be viewed as a minimization condition, not for the energy of a system $\langle\Psi|\hat{{\bf H}}|\Psi\rangle$, but for its quantum “internal energy”, which is

[TABLE]

where $T_{\Psi}$ is referred as the EH or quantum temperature from now on; without loss of generality, one can assume it to be counted with respect to some reference value. The standard thermodynamic arguments yield:

[TABLE]

where the conventional thermodynamic notations are used. In the right-hand side of Eq. (6), the term $\langle\Psi|\hat{{\bf H}}|\Psi\rangle$ comes from the system’s dynamics, while the other term determines the cost of the energy needed to obtain and handle information about a system

[TABLE]

These entropy considerations result in uncertainty relations which can complement the Heisenberg relation ever55 ; hir57 ; bab61 ; beck75 . The physical meaning of the EH conjugate temperature is discussed in the following sections, where it is related to conventional (thermal) temperature, whereas the information-handling cost of energy $\Delta T_{\Psi}S_{\Psi}$ lays quantum-mechanical foundations for the Landauer’s principle.

Yet another interpretation of the coupling $b$ comes from the irreversible dynamics described by the Langevin equation. According to Ref. gg7 , in semiclassical approximation this coupling is related to the friction coefficient. Since the latter is essentially a macroscopic notion, it is unclear whether it has a well-defined analogue in the quantum picture of reality. Therefore, in what follows we will focus on an interpretation of the nonlinear coupling and logarithmic term in terms of $T_{\Psi}$ and EH entropy.

Third, while it appears that the Everett-Hirschman entropy’s considerations are an important step towards a better understanding of the temperature $\Delta T_{\Psi}$, we also must consider what the laws governing its dynamical behavior could be?

These three questions are the main subject of this study. The paper is organized as follows. In Sec. II, we enumerate different ways of deriving wave equations with logarithmic nonlinearity, then we focus on statistical mechanical arguments, and derive a relation between nonlinear coupling and temperature. In Sec. III, we present a model where nonlinear coupling (hence temperature) becomes a dynamical value and introduce the basic notions and equations which we will use in what follows. In Sec. IV, we analytically study some properties and solutions of the model. Discussion and conclusions are presented in Sec. V, where we also outline possible directions for future research.

II Foundations

A wave equation with logarithmic nonlinearity can be introduced into physics using different independent approaches: dilatation covariance ros69 ; ros69a , nonlinear generalization of quantum mechanics preserving energy additivity gg1 ; gg1a , classical Korteweg fluids gg5 ; gg5a ; gl08 ; gl08a , irreversible Langevin dynamics gg7 , measurement and information entropy gg9 , effective nonlinearities in quantum systems zrz17 , superfluidity of helium-4 zlo12 , theory of physical vacuum, quantum gravity and superfluid-gravity correspondence Zloshchastiev:2009zw ; gg15 ; szm16 , to mention only the examples known to the author. In this section, we will present another way in which this equation can manifest, one underlying some of the above-mentioned approaches. This new method of deriving the logarithmic nonlinearity will be based on physically plausible arguments applicable to a large class of many-body systems.

Let us consider a many-body system of particles, whose average potential energy is larger than its kinetic (examples would be systems made of strongly interacting particles, or materials with suppressed kinetic degrees of freedom, such as cold Bose liquids zlo12 or melted thermal insulators in capillary tubes gg5 ). Then the probability of a microstate is given by a Boltzmann rule, in which kinetic energy can be neglected in the leading approximation:

[TABLE]

where $T$, ${\cal E}$ and ${\rm U}$ are, respectively, the temperature, energy and potential energy of a many-body system.

Generally, such a system is described by a large number of linear Schrödinger equations; however, collective degrees of freedom are known to occur in many systems of this kind, which can substantially simplify the theory ps04 . Therefore, if we want to effectively describe our system by a single equation, we must associate the probability $P$ with the wavefunction $\Psi$, which describes collective degrees of freedom, and take into account this statistical effect upon energy of a system. Therefore, we assume $|\Psi|^{2}\sim P\sim\exp{(-{\rm U}/T)}$, from which a general expression for the operator of potential ${\rm U}$ follows: $\hat{\rm U}=-K(T-T_{0})\ln{(A|\Psi|^{2})},$ where $T_{0}$ is a reference temperature, and $A$ and $K$ are some scale constants. Thus, in a position representation, one must include an additional term,

[TABLE]

into the potential part of the initially linear evolution equation for our system. If the system is localized inside a vessel or external potential $V_{\text{ext}}$, then a corresponding term $V_{\text{ext}}\Psi(\vec{r},t)$ must also be added to a wave equation.

For quantum Hamiltonian systems, this wave equation can be written in a standard way:

[TABLE]

where $\hat{\rm H}=i\hbar\partial_{t}$, $\hat{\vec{p}}=-i\hbar\vec{\nabla}$ is a momentum operator in a position representation, and $m$ is an effective mass of a system’s collective degree of freedom.

Finally, after redefining proportionality coefficients, Eqs. (10) and (11) bring us to Eq. (1) where

[TABLE]

where we also recalled the relation (7). This formula indicates that nonlinear coupling is not a fundamental constant, but a dynamical value related to physical observables, such as temperature. Since the latter can generally be a function of position and time, this justifies the necessity of studying logarithmic models with a variable $b$, which will be done in subsequent sections.

A final remark can be added here about the other popular wave equation in the theory of Bose condensates, the Gross-Pitaevskii (GP) equation gp61 ; gp61a , sometimes referred as the cubic Schrödinger equation, related to the Ginzburg-Landau theory. This equation, as well as other higher-order polynomial nonlinear Schrödinger equations arising in the theory of condensed Bose systems, is merely one of perturbative limits of the logarithmic quantum wave equation. This can be demonstrated by expanding a variational functional corresponding to Eq. (1) into the Taylor series in the vicinity of its potential’s nontrivial extrema, see Refs. gg15 ; az11 for details. Indeed, the Taylor series expansion of a logarithmic term in the vicinity of $|\Psi_{\text{ext}}|=a^{-3/2}$ yields

[TABLE]

where the leading-order term can be recognized as the cubic or Gross-Pitaevskii nonlinearity. One can also show that the GP equation describes a special case of dilute Bose-Einstein condensates where the interparticle interaction potential can be approximated by a two-body potential of a contact (delta-singular) shape ps04 .

III The model

Given that we want to upgrade the nonlinear coupling $b$ (hence temperature, according to the previous section) to a dynamical value, we introduce an auxiliary field $\sigma=\sigma(\vec{r},t)$, and define the coupling as its scalar function: $b=b(\sigma)$. Due to the expected gauge invariance of the model, we assume that the resulting wave equation must depend not on the field $\sigma$ itself, but on its derivative such as the gradient $\vec{\nabla}\sigma$. Since the latter is a vector, whereas the coupling must be a scalar function, we assume

[TABLE]

where $\vec{n}=\vec{r}/r=\vec{r}/\sqrt{\vec{r}\cdot\vec{r}}$ is a normal radius vector, and $r=\sqrt{\vec{r}\cdot\vec{r}}$ is an absolute value of the radius vector.

III.1 Minimal model

With assumptions (14) in hand, we keep only the terms which are linear with respect to $\vec{\nabla}\sigma$. Thus, we introduce the simplest (‘minimal’) variable-coupling model as

[TABLE]

where $\rho_{\sigma}=\rho_{\sigma}(\vec{r},t)$ is the energy density of the field $\sigma$, and $\kappa$ is a scale constant. It is natural to assume that this field’s distribution is correlated with our system, therefore, we can impose

[TABLE]

where $f$ is a function which must be specified according to a particular model’s choice, and $\rho=|\Psi|^{2}$ is the probability density of our system; in the case of many-body systems ($N\gg 1$), $\rho$ would be an actual particle density. The exact form of the function $f$ is generally unknown, and the resulting model (15)-(17) is not only nonlinear but also coupled, therefore further analytical studies could become complicated. Fortunately, some robust simplifications can be made in order to extract essential physical information.

Henceforth, we focus on the case of a trapless system, $V_{\text{ext}}\equiv 0$, therefore, one can assume a spatial isotropy of the auxiliary field. We thus set

[TABLE]

hence Eqs. (15) and (16) can be rewritten as:

[TABLE]

where $\nabla^{2}_{r}=\partial^{2}_{r,r}+\frac{2}{r}\partial_{r}$ is a radial part of Laplacian. These two equations must be supplemented with the normalization condition for $\Psi$, boundary conditions for both $\Psi$ and $\sigma$, and a specific expression for a function $f$ depending on the physical system in question.

III.2 Approximate minimal model

The minimal model (19), (20) contains the function $f$ whose value must be specified depending on the physical system in question, otherwise it is generally unknown. However, a certain class of dynamical systems must have a common $f$, at least in the leading approximation.

For simplicity, let us impose here the time independence of the auxiliary field: $\sigma=\sigma(r)$. Furthermore, let us assume that our system satisfies the condition: $\lim\limits_{r\to+\infty}\rho_{\sigma}=\lim\limits_{r\to+\infty}|\Psi|^{2}=0$, therefore its density can be formally represented as a decomposition of the Dirac delta-singular part and asymptotically vanishing extended part:

[TABLE]

where $\delta(r)$ is the Dirac’s delta function centered in the origin. This expression can be viewed as describing a point-like object’s density, plus an extended object’s density distribution represented by a Taylor series expansion with respect to $1/r$, which ensures that the field $\sigma$ vanishes at spatial infinity.

Under these assumptions, Eq. (20) can be approximately written in series form and decoupled from Eq. (19):

[TABLE]

where $Q_{\sigma}=-\tfrac{1}{4\pi}\oint\vec{\nabla}\sigma\cdot d\vec{S}$ and $b_{0}$ are constants (the former being a Gauss law’s charge), and notation ${\cal O}(1/r^{2})$ represents terms which decay faster than $1/r$ when $r\to\infty$. Here, the constant $Q_{\sigma}$ labels the delta-singular part of the density $\rho$, whereas the series coefficient $b_{0}$ labels the leading-order term of the extended part; the chosen notation $b_{0}$ is not a coincidence, as we will see below.

Notice that in the original model (19)-(20), both $Q_{\sigma}$ and $b_{0}$ would not be built-in parameters of a theory, but integration constants, therefore their values would depend on boundary conditions, and could therefore vary from system to system. Thus, the full model would allow us to reduce the number of parameters of a logarithmically nonlinear theory and make it more self-contained. However, within the frameworks of the approximation (22), values of $Q_{\sigma}$ and $b_{0}$ are unknown, and have yet to be determined from other factors.

Furthermore, neglecting higher-order terms ${\cal O}(1/r^{2})$, we can exactly solve Eq. (22). We obtain

[TABLE]

where $\sigma_{0}$ is an additive constant, which can be set to zero due to the gauge invariance of the $\sigma$-field. Substituting this into Eq. (19), we obtain,

[TABLE]

thus confirming our earlier expectations that the nonlinear coupling is not generally constant. If the $\sigma$ field’s charge $Q_{\sigma}$ is nonzero then at $r\to 0$ the coupling’s magnitude grows like $1/r^{2}$, whereas at large $r\to\infty$ the coupling tends to a constant, so that one asymptotically recovers Eq. (4).

Notice that the constant part of the coupling, $b_{0}$, is induced not by the delta-singular part of the $\sigma$-field’s distribution but by its extended part. This explains why solutions of the conventional logarithmic equation (4) are applicable for describing non-singular extended objects, such as Q-balls and finite-size particles em98 ; hkt10 ; Dzhunushaliev:2012zb ; gul14 ; gul15 ; dmz15 and superfluid droplets az11 ; zlo12 . Additionally, the appearance of a new term, proportional to $Q_{\sigma}$, indicates that the new model could also be instrumental in dealing with singular or point-like objects.

For the calculations that follow, we will make Eq. (24) dimensionless. This equation always contains three constants, $\hbar$, $m$ and $a$, which are independent of the nonlinear coupling $b$. Therefore, from them one can construct the following scales of length, time and mass, respectively: $a$, $ma^{2}/\hbar$, $m$. Assuming $a>0$ and

[TABLE]

where $\tau=2ma^{2}/\hbar$, we can write Eq. (24) in a dimensionless form:

[TABLE]

where $\tilde{b}_{0}=b_{0}\tau/\hbar=2mb_{0}a^{2}/\hbar^{2}$ and $\tilde{q}=Q_{\sigma}\tau/(\hbar a^{2})=2mQ_{\sigma}/\hbar^{2}$. In the following sections we will omit primes, assuming that times, lengths, momenta and energies are measured in units of $\tau$, $a$, $\hbar/a$ and $\hbar/\tau$, respectively.

IV Properties and solutions

In this section, we consider a stationary case and analytically derive corresponding solutions. We begin by imposing a stationary ansatz

[TABLE]

where $\tilde{\omega}$ is a frequency measured in units of $1/\tau$. Then Eq. (26) becomes an eigenvalue equation for this frequency:

[TABLE]

where $\psi=\psi(\vec{r})$ is a spatial wavefunction normalized to a number $N$: $\int_{\cal V}|\psi|^{2}d^{3}\vec{r}=N\geqslant 1$.

Due to the symmetry of Eq. (28), it is convenient to work in spherical coordinates from now on. Then the Laplacian can be decomposed into its radial and angular parts

[TABLE]

where $\nabla^{2}_{S^{2}}=\tfrac{1}{\sin\theta}\partial_{\theta}\left(\sin\theta\,\partial_{\theta}\right)+\tfrac{1}{\,\text{sin}^{2}\theta}\partial^{2}_{\varphi,\varphi}$ is the Laplace-Beltrami operator on a sphere.

Here we also introduce the notion of effective external potential. Once a solution of Eq. (28) is known, the effective external potential for such a solution is given, in a dimensionless form, by the expression:

[TABLE]

where $\psi_{s}(\vec{r})$ is a solution’s wavefunction. This potential indicates that a solution $\psi_{s}$ can be equivalently derived from the linear Schrödinger equation with external potential $V_{\text{ext}}=V_{\text{eff}}(\vec{r})$. In other words, an observer would not be able to empirically differentiate a nonlinear problem from a linear one, if based on the analysis of a solution $\psi_{s}$ alone. Note that the effective potential’s shape would vary from solution to solution for the same system, therefore, a nonlinear theory of type (19), (20) has the capacity to describe linear systems with shape-changing external potentials depending on a state, e.g., those systems which undergo phase transitions as their temperature changes. Besides, this creates a framework for creating quantum-mechanical models where an external potential is not ab initio postulated but actually derived.

Another instrumental value to be introduced is a radial density of the Everett-Hirschman entropy (5), measured in units $1/a$:

[TABLE]

where the integral is taken over the sphere; then the entropy (5) is simply $S_{\Psi}=\int_{0}^{\infty}s_{\Psi}^{\text{(r)}}dr$. The $S_{\Psi}$-conjugate temperature in this case is:

[TABLE]

when written in our units of energy $\hbar/\tau$.

Furthermore, when dealing with analytical solutions of Eq. (28), one must distinguish between different cases of nonlinear couplings’ parameters that occur:

IV.1 Case $\tilde{b}_{0}\not=0$,

$\tilde{q}\not=0,1$

In this case, the normalized spherically-symmetric solution of Eq. (28) can be written as:

[TABLE]

and both $\tilde{\omega}$ and coupling $\tilde{b}_{0}$ are no longer arbitrary, but become eigenvalues:

[TABLE]

where the subscript ‘(0)’ denotes the ground state.

Equations (34) and (35) indicate that for this solution to exist, the original parameters $m$, $b_{0}$ and $a$ must not be independent, but must obey a constraint $2mb_{0}a^{2}=\pi\hbar^{2}/N^{2/3}$ instead. For the model’s applications, this can be helpful because it decreases the number of free parameters. Notice also that the (eigen)value of $b_{0}$ depends on the combination of other parameters, namely $ma^{2}N^{2/3}$, which could explain empirical non-observability of logarithmic nonlinear effects in some systems and their dominance in others; further discussion of this can be found in the concluding section.

For the solution (33)-(35), the effective external potential (30) can be evaluated as

[TABLE]

where $\tilde{\Omega}_{\text{eff}}=2\tilde{b}_{0(0)}=2\pi/N^{2/3}$ and $\tilde{q}_{N}=\tilde{q}/\tilde{b}_{0(0)}=\tilde{q}N^{2/3}/\pi$. This formula indicates that most of physical properties of the solution (33)-(35) must be identical to those of a quantum harmonic oscillator of the dimensionless frequency $\tilde{\Omega}_{\text{eff}}$. This correspondence between LogSE’s ground states and quantum harmonic oscillators was noticed in Ref. zrz17 . However, excited states are not likely to be interpreted in terms of the oscillator (36), because the corresponding expression for an effective external potential would certainly be more complex.

Furthermore, the entropy density (31) for the solution (33)-(35) appears to be

[TABLE]

and the integral Everett-Hirschman entropy (5) is simply

[TABLE]

which results in the following relation between $S_{\Psi}$ and frequency’s eigenvalue for this solution which does not contain the normalization number: $\tilde{\omega}_{(0)}S_{\Psi}^{2/3}=\pi\left(3/2\right)^{2/3}(3-\tilde{q})$, where the frequency $\tilde{\omega}_{(0)}$ is defined in Eq. (34).

The quantum temperature (32) becomes in this case:

[TABLE]

The value $\Delta\tilde{T}_{\Psi}$ is always positive-definite if $\tilde{q}\leqslant 0$ which corresponds to a non-negative value of the charge $Q_{\sigma}$. If $\tilde{q}>0$ then the sign of $\Delta\tilde{T}_{\Psi}$ changes when crossing the radius $\sqrt{\tilde{q}_{N}}$ and becomes negative at $r<\sqrt{\tilde{q}_{N}}$. Possible reasons for, and implications of this behavior are discussed in the concluding section.

IV.2 Case $\tilde{b}_{0}\not=0$, $\tilde{q}=1$

In this case, the normalized solution of Eq. (28) and a corresponding frequency eigenvalue can be written as, respectively:

[TABLE]

where the constant $\tilde{k}$ is a solution of the transcendental equation

[TABLE]

while $\tilde{b}_{0}$ remains a free parameter.

For the solution (40), the effective external potential (30) becomes

[TABLE]

thus indicating that most of physical properties of the system in a ground state would be identical to those of a particle trapped in the harmonic potential of a frequency $2\tilde{b}_{0}$ crossed with the Coulomb-type potential with a strength constant $2\tilde{k}$. Notice also the change of the effective potential’s shape compared to the previous case (36) where it is purely harmonic.

The entropy density (31) for the solution (40) can be evaluated as

[TABLE]

and the Everett-Hirschman entropy is

[TABLE]

The quantum temperature (32) becomes in this case:

[TABLE]

thus, similarly to the previous case, $\Delta\tilde{T}_{\Psi}$ is positive at large $r$, changing sign when crossing the radius $\sqrt{\tilde{b}_{0}}$, and becomes negative at small $r$.

IV.3 Case $\tilde{b}_{0}\not=0$, $\tilde{q}=0$

In this case, we recover the logarithmic Schrödinger equation with a constant nonlinear coupling,

[TABLE]

discussed after Eq. (4) above.

For a stationary case (27), the normalized spherically-symmetric ground-state solution of Eq. (47) and a corresponding frequency eigenvalue can be written as, respectively:

[TABLE]

which describes a Gaussian-shaped spherical wave.

For the solution (48), the effective external potential (30) reads

[TABLE]

which makes this case similar to Eq. (36): most of physical properties of the solution (48) must be determined by a quantum harmonic oscillator of the mass $1/2$ and dimensionless frequency $2\tilde{b}_{0}$.

The entropy density (31) for this solution appears to be

[TABLE]

hence the integrated Everett-Hirschman entropy is simply

[TABLE]

where the frequency $\tilde{\omega}_{(0)}$ is defined in Eq. (49). This results in the following relation between $S_{\Psi}$ and frequency’s eigenvalue for this solution:

[TABLE]

The conjugate quantum temperature (32) becomes a constant in this case

[TABLE]

which is positive-definite.

IV.4 Case $\tilde{b}_{0}=0$, $\tilde{q}\not=0$

In this case, Eq. (26) becomes

[TABLE]

where a decomposition (29) is implied.

While this equation looks more complicated to solve than the logarithmic equation with a constant nonlinear coupling (47), it has certain features which make it easier to study.

Most significantly, this equation allows a separation of angular variables from others, which paves the way for us to drastically decrease the dimensionality of the problem in a general case. Assuming the stationary ansatz (27), where

[TABLE]

and using the decomposition (29), we can separate Eq. (55) into its radial and angular parts:

[TABLE]

where $L$ is a separation constant whose eigenvalue follows from the last equation. The latter resembles a differential equation for spherical harmonics, but contains a nonlinear term, making its solutions a separate topic of research. This equation indicates that the system’s total angular momentum acquires a nonlinear correction which can manifest in those systems for which $Q_{\sigma}\not=0$.

Furthermore, equation (55) can be viewed either as a short-distance limit $r\ll\sqrt{\tilde{q}/\tilde{b}_{0}}=Q_{\sigma}/(b_{0}a^{2})$ of Eq. (26), or as a large-charge limit, $Q_{\sigma}\to\infty$, thereof. It is thus no longer necessary to assume that $b_{0}$ is unnaturally small to fit existing experimental data for physical systems for which the model (4) is a priori inapplicable. It is sufficient to assume that models with a large value of $Q_{\sigma}$ are more relevant for those systems.

Because the separation of angular variables from others is possible in this case, we need not restrict ourselves to a spherically symmetric ansatz to find analytical solutions. Instead, we will search for a solution for the radial wavefunction $R(r)$ given by Eq. (57) and normalized as $\int_{0}^{\infty}|R|^{2}r^{2}dr=N$. Then the solution of Eq. (57) can be written as:

[TABLE]

where $\tilde{\mu}^{2}=(4N)^{-1/3}\exp{(-L^{2}/3)}$, and a subscript ‘0’ refers to a lowest value of a main quantum number. It is clear that on a real semi-axis a physically suitable solution is the one for which

[TABLE]

which can be regarded as a constraint for the original value $Q_{\sigma}$: $Q_{\sigma}\to\hbar^{2}/(2m)$.

For the solution (61)-(65), the effective external potential (30) reads:

[TABLE]

thus indicating that most of physical properties of this solution would be identical to those of a particle in the Coulomb-type potential with a strength constant $2\tilde{\mu}^{2}$, in presence of the standard “centrifugal” potential $L^{2}/r^{2}$.

Furthermore, the entropy density (31) for the solution (61)-(65) appears to be

[TABLE]

where we denoted the constant

[TABLE]

an integration being taken over a sphere. The integral EH entropy (5) reads:

[TABLE]

while the conjugate quantum temperature (32) becomes in this case simply:

[TABLE]

implications of which are discussed below.

V Discussion

We have studied the dynamical behavior of nonlinear coupling $b$ in the quantum wave equation of a logarithmic type. Using statistical mechanics arguments, we have shown that this coupling is related to the thermal temperature of many-body systems which satisfy the following conditions: (i) their potential energy must be larger than kinetic, (ii) they must allow an effective description in terms of collective degrees of freedom governed by a wavefunction. One example of such systems would be strongly interacting Bose liquids, where the logarithmic model was shown to be very instrumental az11 ; zlo12 .

Furthermore, the nonlinear coupling has been associated with a certain kind of quantum temperature: a thermodynamical conjugate to the Everett-Hirschman entropy, which allows us to relate thermal temperature, EH quantum temperature, and nonlinear coupling, as discussed in Sec. II.

In view of the dynamical nature of nonlinear coupling, a combined quantum-mechanical and field-theoretical model is proposed in Sec. III, which leads to a logarithmic equation with variable nonlinear coupling. By studying the behavior of the latter, one achieves deeper understanding of the Everett-Hirschman entropy and its thermodynamical conjugate. Considering this model in a next-to-leading order approximation with respect to the auxiliary field $\sigma$, see Sec. III.2, we analytically obtained a number of stationary solutions and established a number of profound properties, given in Sec. IV. Below we present discussion of the results obtained.

First, the logarithmic model reveals a connection between the EH conjugate temperature and the conventional (thermal) one. The latter is still a notion which is well-defined at a classical level only (e.g., as measured by bringing a system into a thermal equilibrium with a calibrated system), but its quantum analogue is unknown. This problem manifests itself most strikingly when dealing with cold quantum gases and liquids such as Bose-Einstein condensates. Strictly speaking, one can not measure the temperature of a condensate in a classical way – experimental condensates are energetically isolated, so no thermal equilibrium can be achieved with a calibrated system without affecting the condensate’s state. Thus, the standard method of measurement consists of switching a trap off, using a laser upon the condensate’s atoms and measuring the scattered light to deduce their temperature from experimental profiles of density and momentum distributions, assuming that the energy spectrum is also known. However, this method of measurement presumes that relations between temperature and the above-mentioned distributions and spectrum are derived from some theoretical model, which must be thus presumed to be a priori valid for that particular condensate. Besides, the detrapping measurement is unlikely to be reliable for strongly interacting quantum Bose liquids, which tend to confine their atoms into a droplet, in absence of trapping potentials az11 ; zlo12 ; z17zna .

In this regard, a conjecture that thermal temperature is related to quantum temperature $T_{\Psi}$ and nonlinear coupling $b$, at least for a large class of systems, has been discussed in Sec. II. Aside from solving the above-mentioned issue with the fundamental (quantum-mechanical) definition of the thermal temperature, the relation (12) lays quantum-mechanical foundations also for the Landauer’s principle lan61 . Namely, it is the term $\Delta T_{\Psi}\,S_{\Psi}$ from Eq. (6) which is responsible for information-handling cost of energy, including the energy cost for information erase. The latter has been confirmed in experiments with different nanoscale systems bap12 ; jgb14 ; hld16 .

Second, in the model with variable nonlinear coupling, the nature of quantum EH temperature (6) becomes clearer. For the solution described in Sec. IV.3, the value $\Delta T_{\Psi}$ is positive semi-definite, whereas in Sec. IV.4 it is negative semi-definite; in Secs. IV.1 and IV.2 it changes its sign at a certain value of radius. The common feature of the cases in Secs. IV.1 and IV.2 is that $\Delta T_{\Psi}$ tends toward negative values at small distances from the origin and to a positive constant at large distances. Moreover, the common feature of all cases is that $\Delta T_{\Psi}$ tends to negative values whenever the term $Q_{\sigma}/r^{2}$ in the nonlinear coupling predominates over the constant one. Analyzing these features together, one can hypothesize that the EH temperature can serve as a means of differentiating phases, e.g., those related to the microscopic and macroscopic scales of radius: $\Delta T_{\Psi}$ is negative for the microscopic scale and it is positive for the macroscopic one (up to a sign convention adopted in the definition of $S_{\Psi}$).

Third, in the minimal model of Sec. III, the coupling constant $b_{0}$ is no longer a predefined parameter of a theory, cf. Eq. (4). Instead, it becomes one of the integration constants of evolution equations, such as Eqs. (19) and (20), therefore, its value can vary from system to system. Thus, the full model allows us to reduce a number of parameters of the theory and make it more self-consistent and self-sufficient. However, within the frameworks of the approximation (22)-(24), values of $Q_{\sigma}$ and $b_{0}$ are unknown and have yet to be determined from other considerations.

Nevertheless, even the approximate minimal model, cf. Sec. III.2, offers an explanation as to why the constant $b_{0}$ is negligible for some systems but crucial for others, as mentioned in the Introduction. To illustrate this, let us compare cases described in Sec. IV. For the solutions of Eq. (24), which are described in Secs. IV.2 and IV.3, the constant $b_{0}$ remains a free parameter, which can take any value, either defined ad hoc or fitted from an experiment; an example of the latter procedure can be found in Ref. zlo12 . However, for the solution in Sec. IV.1, the constant $b_{0}$ becomes an eigenvalue, i.e., a function of other constants of the model and quantum numbers (if one considers excited states). Specifically, it is small if the combination $ma^{2}N^{2/3}$ is large. Because neither of those three constants are fundamental nor universal for all quantum systems, the value of $b_{0}$ can vary between systems. Moreover, it can also vary for different solutions of the same system, because of the above-mentioned eigenvalue structure and a Hilbert space associated with it.

Finally, effective external potentials computed in Secs. IV.2 and IV.4 illustrate a possibility that some fundamental interactions, such as gravity, could emerge as a nonlinear quantum-mechanical phenomenon based on a concept of the quantum information entropy, cf. Eq. (6), and evolution equations of a logarithmic type. This conjecture is supported by other studies, which suggest that the most probable candidate for such a phenomenon is a background superfluid of a logarithmic type Zloshchastiev:2009zw ; gg15 ; szm16 .

Acknowledgements.

Discussions with I. Sinaysky from University of Kwazulu Natal‎ (who brought the Landauer’s work into my attention) and A. Avdeenkov from I.I. Leypunsky Institute of Physics and Power Engineering are acknowledged. Proofreading of the manuscript by P. Stannard is greatly appreciated. This work is based on the research supported by the National Research Foundation of South Africa under Grants Nos. 95965 and 98892.