# Classical stochastic approach to quantum mechanics and quantum
thermodynamics

Mário J. de Oliveira  Universidade de São Paulo, Instituto de Física, Rua do
Matão, 1371, 05508-090 São Paulo, SP, Brazil

###### Abstract

We derive the equations of quantum mechanics and quantum thermodynamics from
the assumption that a quantum system can be described by an underlying
classical system of particles. Each component $\phi_{j}$ of the wave vector is
understood as a stochastic complex variable whose real and imaginary parts are
proportional to the coordinate and momentum associated to a degree of freedom
of the underlying classical system. From the classical stochastic equations of
motion, we derive a general equation for the covariance matrix of the wave
vector which turns out to be of the Lindblad type. When the noise changes only
the phase of $\phi_{j}$, the Schrödinger and the quantum Liouville equation
are obtained. The component $\psi_{j}$ of the wave vector obeying the
Schrödinger equation is related to stochastic wave vector by
$|\psi_{j}|^{2}=\langle|\phi_{j}|^{2}\rangle$.

##  I Introduction

A distinguishing feature of quantum mechanics landau1958 ; merzbacher1961 ;
messiah1961 ; sakurai1967 ; sakurai1994 ; griffiths1995 ; piza2002 ;
griffiths2002  is its formulation in terms of an unobservable: the wave
function. The presence of unobservables in a theory does not make it
unscientific, as long as they lead to observables, that is, to quantities that
can be observed or measured experimentally oliveira2021 . Any scientific
theory has unobservables to a greater or lesser extent. Significant examples
are the epicycles of Ptolemy, the aether of Newton, and the luminiferous
aether of Maxwell. Much less obvious examples are the concepts of time
reversibility, causality, homogeneity of time and homogeneity of space. The
unobservable of quantum mechanics was present in its very beginning when
Schrödinger formulated his equation in terms of the wave function. Although
the wave function is an unobservable it leads through this equation to the
observable spectral lines of hydrogen.

The question that we address here concerns the possibility of the formulation
of quantum mechanics an unobservable other than the wave function.
Specifically, the unobservable that we consider here to describe a quantum
system is a system of particles obeying the classical equations of motion
lanczos1949 ; goldstein1950 ; landau1960 ; arnold1978 , which we call the
underlying system to avoid confusion with a real system described by classical
mechanics.

The standard formulation of quantum mechanics postulates that the quantum
states are represented by wave vectors belonging to a complex vector space,
the Hilbert space. In contrast, classical mechanics is represented in the
Hamilton formulation by canonical variables belonging to a real vector space,
the phase space. Thus a classical approach to quantum mechanics, requires a
formulation of classical mechanics in terms of complex canonical variables.

The possibility of using complex variables to express the classical Hamilton
equations of motion was pointed out by Lanczos lanczos1949  who showed that a
pair of complex conjugate variables is also a pair of canonical variables. The
formulation of the quantum equation of motion by a classical Hamilton equation
in complex variables was given by Strocchi strocchi1966 . In his formulation a
complex canonical variable is identified as a component $\phi_{j}$, which
obeys the complex Hamilton equation associated to the Hamiltonian function
${\cal H}$, identified as the mean value

${\cal H}=\sum_{j}\phi_{j}^{*}H_{jk}\phi_{k}$ (1)

of the Hamilton operator $H$.

An equivalent approach was proposed by Heslot heslot1985  but instead of using
a complex Hamilton equation, he shows that the real and imaginary parts of the
wave functions are a pair of real classical canonical variables obeying the
real standard Hamilton equation. The classical representation was then
analyzed and explored by several authors gray1994 ; bodurov1998 ; briggs2012 ;
elze2012 .

A essential aspect of the classical representation concerns the norm of the
wave vector,

${\cal N}=\sum_{j}\phi_{j}^{*}\phi_{j},$ (2)

which is a quantity conserved by the Hamilton equations of motion associated
to the classical Hamiltonian (1). This is a nice property since according the
wave vector should be normalized at all times. However, the conservation
property does not determine the value of the norm. Therefore we should
postulate that the norm has the same value for any possible trajectory in
phase space as this is a basic postulate of quantum mechanics. More precisely,
among all sectors of the phase space determined by distinct values of the
norm, we must select just one of them. As we shall see, the one to be selected
is connected to the Planck constant.

Quantum mechanics is understood as having a probabilistic character. However
this character is not clearly manifested in the usual representation. For
instance, no variable is considered to be a random variable. The probabilistic
character is a consequence of the standard interpretation of quantum mechanics
omnes1994 ; auletta2001 ; freire2022  which introduces probability in an ad-
hoc and a posteriori manner by the proposition that the square of the absolute
value of the wave function is a probability.

Here we introduce the probabilistic character in an explicit and a priori form
by turnig $\phi_{j}$ into a stochastic variable. This is accomplished by
transforming the equation of motion into a stochastic equation through the
addition of a noise term in the Hamilton equations of motion vankampen1981 ;
risken1989 ; gardiner2009 ; tome2015 . The noise transforms the trajectories
in the complex phase space into stochastic trajectories, and it is set up in
such a way as to preserve the norm given by (2) along any stochastic
trajectory.

The noise changes in general the absolute value and the phase of $\phi_{k}$. A
meaningful result of our analysis is that the Schrödinger and the quantum
Liouville equations are obtained through noises that change the phase but not
the absolute value of $\phi_{k}$. This type of noise also makes each term of
the norm constant. If the noise changes both the phase and the absolute value
of $\phi_{k}$ then we reach the quantum thermodynamic equation which turns out
to be the Lindblad equation for open system breuer2002 ; manzano2020 ;
lindblad1976 ; gorini1976 .

The quantum thermodynamic equation is the central equation of the quantum
thermodynamics that we develop here. Based on this equation we derive the
first law of thermodynamics and the second law of thermodynamics. This is
accomplished by defining the quantum entropy and the quantum entropy
production. The second law of thermodynamics is obtained by demonstrating that
the entropy production is nonnegative. To demonstrate this proposition we use
a reasoning similar to that employed by Spohn spohn1978  which in turn was
based on a theorem of Lieb lieb1973 .

##  II Stochastic equation of motion

###  II.1 Underlying classical system

The underlying system consists of a classical system with $n$ degrees of
freedom. Each degree of freedom corresponds to a pair of canonical variables
$q_{i}$ and $p_{i}$, and the Hamilton equations of motion are

$\frac{dq_{i}}{dt}=\frac{\partial{\cal H}}{\partial
p_{i}},\qquad\frac{dp_{i}}{dt}=-\frac{\partial{\cal H}}{\partial q_{i}},$ (3)

which determine trajectories in the $2n$ dimensional phase space, which is the
vector space spanned by the canonical variables. The Hamiltonian function is
assumed to be of the form

${\cal H}=\frac{1}{2m}\sum_{j}p_{j}^{2}+\frac{1}{2}\sum_{jk}K_{jk}q_{j}q_{k},$
(4)

which corresponds to a collection of $n$ particles of mass $m$ interacting
harmonically. The coefficients $K_{jk}$ are the entries of a $n\times n$
matrix $K$ with positive eigenvalues. Using the notations $q$ and $p$ for the
column matrices with elements $q_{j}$ and $p_{j}$, respectively, we write

${\cal H}=\frac{1}{2m}p^{\\!\sf T}p+\frac{1}{2}q^{\\!\sf T}Kq.$ (5)

We perform a canonical transformation to write the Hamiltonian function in a
more symmetric form in which it becomes invariant by the exchange of the
coordinate and momentum of a pair of canonically conjugate variables. Taking
into account that $K$ is Hermitian with nonnegative eigenvalues we may define
the matrix $\Omega=K^{1/2}/\sqrt{m}$. It has nonnegative eigenvalues and is
Hermitian. The canonical transformation $q\to x$ and $p\to y$ is defined by

$x=(m\Omega)^{1/2}q\qquad y=(m\Omega)^{-1/2}p,$ (6)

and the Hamiltonian function becomes

${\cal H}=\frac{1}{2}y^{\\!\sf T}\Omega y+\frac{1}{2}x^{\\!\sf T}\Omega x,$

and we see that ${\cal H}$ is invariant by the exchange of $x$ and $y$. In an
explicit form,

${\cal H}=\frac{1}{2}\sum_{jk}\Omega_{jk}(x_{j}x_{k}+y_{j}y_{k}),$ (7)

and the Hamilton equations of motion becomes

$\frac{dx_{i}}{dt}=\frac{\partial{\cal H}}{\partial
y_{i}},\qquad\frac{dy_{i}}{dt}=-\frac{\partial{\cal H}}{\partial x_{i}}.$ (8)

Let us define the quantity

${\cal I}=\frac{1}{2}\sum_{j}(x_{j}^{2}+y_{j}^{2}).$ (9)

It follows from the Hamilton equations that ${\cal I}$ is a constant of the
motion, a property that allows us to divide the phase space in sectors, each
one corresponding to a given value of ${\cal I}$. We then postulate that the
only possible motions of the underlying system are those corresponding to a
defined value ${\cal I}$. This value is denoted by $\mu$,

$\frac{1}{2}\sum_{j}(x_{j}^{2}+y_{j}^{2})=\mu.$ (10)

The constant $\mu$ has the physical dimension of (energy$\times$time) and it
will be seen that $\mu$ is to be identified as the Planck constant. This
postulate is the crucial step toward quantization within the present approach.
It enables the appearance in the classical underlying system of the
characteristic quantum properties such as the quantization of energy and the
zero point of energy.

One can show that ${\cal I}$ is an adiabatic invariant oliveira2022 , that is,
a slow change of the parameters of the Hamiltonian will change ${\cal H}$ but
not ${\cal I}$. This result connects the adiabatic invariance with
quantization, an idea advanced by Ehrenfest which lead to the old quantum
mechanics. The distinction between the approach of Ehrenfest from ours lies in
his use of the invariant as connected to the motion of one quantum particle as
if the particle itself was a classical particle.

In fact ${\cal I}$ is not constant only in adiabatic changes but it is
universally constant. Let us suppose that ${\cal H}$ is time dependent, that
is, $\Omega_{jk}$ depends on time. In this case ${\cal H}$ is not a constant
of motion because

$\frac{d{\cal
H}}{dt}=\frac{1}{2}\sum_{jk}\frac{d\Omega_{jk}}{dt}(x_{j}x_{k}+y_{j}y_{k}),$
(11)

but

$\frac{d{\cal I}}{dt}=0$ (12)

This result allows to say that ${\cal I}$ remains forever with the same value
given at the initial time, even if ${\cal H}$ is time dependent.

###  II.2 Classical dynamics in complex variables

The Hamilton equations of motion are written in terms of a set of $2n$ complex
variables $\phi_{j}$ and $\phi_{j}^{*}$, $j=1,\ldots,n$, obtained by the
transformation

$\phi_{j}=\frac{1}{\sqrt{2\mu}}(x_{j}+iy_{j}),\qquad\phi_{j}^{*}=\frac{1}{\sqrt{2\mu}}(x_{j}-iy_{j}).$
(13)

Using this transformation, we see that the expression (10) becomes

$\sum_{j}\phi_{j}^{*}\phi_{j}=1,$ (14)

which expresses the postulate introduced above in terms of the complex
variables.

The peculiar transformation from real to complex variables are canonical
transformation leading to the following Hamilton equation of motion in complex
variables

$\frac{d\phi_{j}}{dt}=\frac{1}{i\mu}\frac{\partial{\cal
H}}{\partial\phi_{j}^{*}},\qquad\frac{d\phi_{j}^{*}}{dt}=-\frac{1}{i\mu}\frac{\partial{\cal
H}}{\partial\phi_{j}},$ (15)

where ${\cal H}$ is the real bilinear function

${\cal H}=\sum_{jk}H_{jk}\phi_{j}^{*}\phi_{k},$ (16)

obtained from (7), where $H_{jk}=\mu(\Omega_{jk}+\Omega_{kj})/2$, and are the
entries of a $n\times n$ Hermitian matrix $H$, that is, $H_{jk}^{*}=H_{kj}$,
with positive eigenvalues. We remark that the complex variables $\phi_{j}$ and
$\phi_{j}^{*}$ are dimensionless, which justifies the presence of the constant
$\mu$ in the equations of motion.

The complex conjugate variables $\phi_{j}$ and $\phi_{j}^{*}$ are considered
to be independent variables because their real and imaginary parts are
proportional to the coordinate $q_{j}$ and to the momentum $p_{j}$, which are
independent. From the peculiar transformation above they also form a pair of
canonically conjugate variables as is manifest in the equation of motion (15).

From the Hamilton equations of motion (15), the time evolution of a state
function ${\cal F}$ is

$\frac{d{\cal F}}{dt}=\\{{\cal F},{\cal H}\\},$ (17)

where the term in the right-hand side is the Poisson brackets defined by

$\\{{\cal F},{\cal H}\\}=\sum_{j}(\frac{\partial{\cal
F}}{\partial\phi_{j}}\frac{\partial{\cal
H}}{\partial\phi_{j}^{*}}-\frac{\partial{\cal
H}}{\partial\phi_{j}}\frac{\partial{\cal F}}{\partial\phi_{j}^{*}}),$ (18)

If we replace $\cal F$ by the norm of $\phi$, defined by

${\cal N}=\sum_{j}\phi_{j}^{*}\phi_{j},$ (19)

we see that

$\\{{\cal N},{\cal H}\\}=0,$ (20)

which means that the norm is preserved along a trajectory in the complex
vector space.

The value of ${\cal N}$ cannot be arbitrary. In accordance with the postulate
introduced above and expressed by equation (14), it should be equal to the
unity. Using this result and writing the axes corresponding to a pair of
canonical conjugate variables of the phase space as related to a complex
variable $\phi_{j}$, then the phase space becomes equivalent to a Hilbert
space, which is a vector space with normalized vectors.

The form (16) of the Hamiltonian allows us to write the Hamilton equations in
the form

$\frac{d\phi_{j}}{dt}=\frac{1}{i\mu}\sum_{k}H_{jk}\phi_{k}.$ (21)

Equivalently, we may write equation (21) in the vector form

$i\mu\frac{d\phi}{dt}=H\phi.$ (22)

If we set $\mu=\hbar$ and identify $\phi$ as the quantum state vector, we see
that it is identical to the quantum equation that gives the time evolution of
the state vector $\phi$, or the Schrödinger equation.

###  II.3 Norm preserving noise

We now assume that $\phi$ follows a stochastic equation of motion which is the
Hamilton equations supplemented by a noise term,

$\frac{d\phi_{j}}{dt}=\frac{1}{i\mu}\sum_{jk}H_{jk}\phi_{k},+\zeta_{j},$ (23)

where $\zeta_{j}$ is a complex white noise, that is, its real and imaginary
parts are white noises, and also depends on $\phi$ and $\phi^{*}$. The noise
will be set up in such a way that the norm ${\cal N}$ is preserved along the
stochastic trajectory in the complex vector space.

As it stands equation (23) has no precise meaning. To give it a precise
meaning, we discretize the time in intervals $\tau$ and write the stochastic
equation of motion in a discretized form. Let $\Delta\phi_{j}$ be the
increment in $\phi_{j}$ when the time increases from $t$ to $t+\tau$. Then the
discretized version of equation (23) is assumed to be

$\Delta\phi_{j}=\frac{\tau}{i\mu}\sum_{k}H_{jk}\phi_{k}+i\sqrt{\tau}\sum_{k}G_{jk}\phi_{k}-\frac{\tau}{2}\sum_{k}K_{jk}\phi_{k},$
(24)

where $G_{jk}$ are independent random variables, which we consider to be the
elements of a $n\times n$ matrix $G$, with zero mean, $\langle
G_{jk}\rangle=0$, and covariances

$\langle G_{jk}^{*}G_{\ell m}\rangle=2\gamma_{jk,\ell m}.$ (25)

from which follows the property $\gamma_{jk,\ell m}^{*}=\gamma_{\ell m,jk}$.
The quantities $K_{jk}$ are related to $G_{jk}$ and will be found by imposing
the conservation of the norm. The elements $G_{jk}$ and $K_{jk}$ are
understood as the entries of two $n\times n$ matrices $G$ and $K$,
respectively.

It is convenient to introduce a $n^{2}\times n^{2}$ matrix $\Gamma$ with
elements $\Gamma_{rs}$ which are related to $\gamma_{jk,\ell,m}$. The index
$r$ has a one-to-one correspondence to $(jk)$ and $s$ has a one-to-one
correspondence to $(\ell m)$. Using this convention, set
$\Gamma_{rs}=\gamma_{r,s}$, or

$\Gamma_{rs}=\langle G_{r}^{*}G_{s}\rangle.$ (26)

From this relation it follows that $\Gamma$ is a Hermitian matrix,
$\Gamma_{rs}^{*}=\Gamma_{sr}$. Being Hermitian, $\Gamma$ can be diagonalized
by a unitary transformation. Denoting by $\Upsilon$ the $n^{2}\times n^{2}$
matrix that diagonalizes $\Gamma$ then $\Upsilon^{\dagger}\Gamma\Upsilon$ is
diagonal and its elements $\lambda_{r}$ are the eigenvalues of $\Gamma$. They
are given by

$\lambda_{r}=\sum_{s^{\prime}s}\Upsilon^{\dagger}_{rs^{\prime}}\Gamma_{s^{\prime}s}\Upsilon_{sr}.$
(27)

Using (26),

$\lambda_{r}=\sum_{s^{\prime}s}\Upsilon^{\dagger}_{rs^{\prime}}\langle
G^{*}_{s^{\prime}}G_{s}\rangle\Upsilon_{sr}=\langle|\sum_{s}G_{s}\Upsilon_{sr}|^{2}\rangle,$
(28)

and we see that $\lambda_{r}\geq 0$, that is the eigenvalues of the Hermitian
matrix $\Gamma$ are nonnegative, or in other words, $\Gamma$ is a positive
semi-definite matrix.

Let us determine the increment in the norm ${\cal N}$ due to a change
$\Delta\phi_{j}$ in the dynamic variables. It is given by

$\Delta{\cal
N}=\sum_{j}(\Delta\phi_{j}^{*}\phi_{j}+\phi_{j}^{*}\Delta\phi_{j}+\Delta\phi_{j}^{*}\Delta\phi_{j}).$
(29)

Replacing $\Delta\phi_{j}$ in this equation, we find up to terms of order
$\tau$

$\Delta{\cal N}=i\sqrt{\tau}\sum_{jk}(G_{jk}-G_{kj}^{*})\phi_{j}^{*}\phi_{k}$
$-\frac{\tau}{2}\sum_{jk}(K_{jk}+K_{kj}^{*})\phi_{j}^{*}\phi_{k}+\tau\sum_{jk}(G^{\dagger}G)_{jk}\phi_{j}^{*}\phi_{k}.$
(30)

The terms containing $H_{jk}$ vanish identically due to the Hermitian property
$H_{jk}^{*}=H_{kj}$. Choosing

$K=G^{\dagger}G,$ (31)

which is the sought relation between $K$ and $G$, and which we assume from now
on, then the last two summations on the right-hand side of (30) vanish and

$\Delta{\cal
N}=i\sqrt{\tau}\sum_{jk}\phi_{j}^{*}(G-G^{\dagger})_{jk}\phi_{k}.$ (32)

If $G^{\dagger}=G$ then the increment vanishes and the norm $\cal N$ is
strictly constant along the stochastic trajectory. If this condition is not
imposed, the increment in the norm will still vanish but in the average, that
is, $\langle\Delta{\cal N}\rangle=0$. The stochastic equation of motion (24)
defines a Markovian stochastic dynamics which determines stochastic
trajectories of $\phi$ in the vector space.

##  III Fundamental equation

###  III.1 Probability density distribution

As the trajectories in the vector space are stochastic, we may ask for the
probability of the occurrence of each trajectory. In the following we
determine the equation that gives the time evolution of the probability
density distribution ${\cal P}(\phi,\phi^{*},t)$ of $\phi$ and $\phi^{*}$ at
time $t$. We start by considering an arbitrary state function ${\cal F}$ of
$\phi$ and $\phi^{*}$ of the bilinear type

${\cal F}=\sum_{jk}F_{jk}\phi_{j}^{*}\phi_{k},$ (33)

where $F_{jk}$ are understood as the complex entries of a $n\times n$ matrix
$F$. The increment $\Delta{\cal F}$ of such a function is given by

$\Delta{\cal F}=\sum_{j}\frac{\partial{\cal
F}}{\partial\phi_{j}}\Delta\phi_{j}+\sum_{j}\frac{\partial{\cal
F}}{\partial\phi_{j}^{*}}\Delta\phi_{j}^{*}$
$+\sum_{jk}\frac{\partial^{2}{\cal
F}}{\partial\phi_{j}\partial\phi_{k}^{*}}\Delta\phi_{j}\Delta\phi_{k}^{*}.$
(34)

Replacing $\Delta\phi_{j}$ in this equation we find up to terms of order
$\tau$

$\Delta{\cal F}=\frac{\tau}{i\mu}\\{{\cal F},{\cal
H}\\}+i\sqrt{\tau}\sum_{jk}(\frac{\partial{\cal
F}}{\partial\phi_{j}}G_{jk}\phi_{k}-\frac{\partial{\cal
F}}{\partial\phi_{k}^{*}}G_{kj}^{*}\phi_{j}^{*})$
$-\frac{\tau}{2}\sum_{jk}(\frac{\partial{\cal
F}}{\partial\phi_{j}}K_{jk}\phi_{k}+\frac{\partial{\cal
F}}{\partial\phi_{k}^{*}}K_{kj}^{*}\phi_{j}^{*})$ $+\tau\sum_{jk\ell
m}\frac{\partial^{2}{\cal
F}}{\partial\phi_{j}\partial\phi_{k}^{*}}G_{j\ell}G_{km}^{*}\phi_{\ell}\phi_{m}^{*}.$
(35)

Taking the average of both sides of equation (35), the term proportional to
$\sqrt{\tau}$ vanishes. After that, we divide what is left by $\tau$ to reach
the result

$\frac{d}{dt}\langle{\cal F}\rangle=\frac{1}{i\mu}\langle\\{{\cal F},{\cal
H}\\}\rangle-\sum_{jk\ell}\gamma_{\ell j,\ell
k}(\langle\phi_{k}\frac{\partial{\cal
F}}{\partial\phi_{j}}\rangle+\langle\phi_{j}^{*}\frac{\partial{\cal
F}}{\partial\phi_{k}^{*}}\rangle)$ $+2\sum_{jk\ell
m}\gamma_{km,j\ell}\langle\phi_{\ell}\phi_{m}^{*}\frac{\partial^{2}{\cal
F}}{\partial\phi_{j}\partial\phi_{k}^{*}}\rangle.$ (36)

where here the average are taken over the probability density distribution
${\cal P}$, that is, the average $\langle{\cal F}\rangle$ of ${\cal F}$ is

$\langle{\cal F}\rangle=\int{\cal F}{\cal P}d\phi d\phi^{*}.$ (37)

Taking into account that ${\cal F}$ is an arbitrary function, we reach the
equation for the time evolution of the probability distribution ${\cal P}$,
which is

$\frac{\partial{\cal P}}{\partial t}=\frac{1}{i\mu}\\{{\cal H},{\cal
P}\\}+\sum_{jk\ell}\gamma_{\ell j,\ell k}(\frac{\partial\phi_{k}{\cal
P}}{\partial\phi_{j}}+\frac{\partial\phi_{j}^{*}{\cal
P}}{\partial\phi_{k}^{*}})$ $+2\sum_{jk\ell
m}\gamma_{km,j\ell}\frac{\partial^{2}\phi_{\ell}\phi_{m}^{*}{\cal
P}}{\partial\phi_{j}\partial\phi_{k}^{*}}.$ (38)

To reach this equation we bear in mind that the averages in (36) are integrals
in the complex vector space of the type (37). The expressions in (38) are
found performing appropriate integrals by parts and considering that ${\cal
P}$ vanishes rapidly at the boundaries of integration. The equation (38) is
recognized as a Fokker-Planck-Kramers equation vankampen1981 ; risken1989 ;
gardiner2009 ; tome2015 , in several complex variables.

###  III.2 Master equation

The fundamental equation (38) was derived above considering a discrete time
stochastic equation of motion and then taking the continuous time limit. Here
we consider another derivation of the fundamental equation by considering a
continuous time equation of motion but discretized variables $\phi$ and
$\phi^{*}$. To simplify the notation we write $\varphi$ in the place of
$(\phi,\phi^{*})$ and consider the following Kolmogorov equation, or master
equation,

$\frac{d}{dt}{\cal
P}(\varphi)=\sum_{\varphi^{\prime}}\\{W(\varphi,\varphi^{\prime}){\cal
P}(\varphi^{\prime})-W(\varphi^{\prime},\varphi){\cal P}(\varphi)\\},$ (39)

where $W(\varphi^{\prime},\varphi)$ are the entries of a stochastic matrix
$W$. A stochastic matrix holds two properties: 1) the off diagonal entries
$W(\varphi^{\prime},\varphi)$ are nonnegative and represent the probability
rate of the transition $\varphi\to\varphi^{\prime}$, and 2) it holds the
property

$\sum_{\varphi}W(\varphi,\varphi^{\prime})=0.$ (40)

From these two properties, it follows from the Perron-Frobenius theorem that
${\cal P}(\varphi,t)\geq 0$ if at the initial time it holds this property, and
it is normalized at all times,

$\sum_{\varphi}{\cal P}(\varphi,t)=1.$ (41)

It is more convenient to construct the backward Kolmogorov equation, or
adjoint master equation,

$\frac{d}{dt}{\cal
Q}(\varphi)=\sum_{\varphi^{\prime}}W(\varphi,\varphi^{\prime})\\{{\cal
Q}(\varphi^{\prime})-{\cal Q}(\varphi)\\}.$ (42)

We consider three types of transitions. The first is defined by

$\phi_{j}\to\phi_{j}^{\prime}=\phi_{j}+\frac{\varepsilon}{i\mu}H_{j\ell}\phi_{\ell},$
(43)
$\phi_{k}^{*}\to\phi_{k}^{\prime*}=\phi_{k}^{*}-\frac{\varepsilon}{i\mu}H_{mk}\phi_{m}^{*},$
(44)

and occurs with rate one. The contribution of this transition to the right-
hand side of equation (42) is given by the expression

$\frac{1}{i\mu}\sum_{j\ell}\\{H_{j\ell}\phi_{\ell}\frac{\partial{\cal
P}}{\partial\phi_{j}}-H_{\ell j}\phi_{\ell}^{*}\frac{\partial{\cal
P}}{\partial\phi_{j}^{*}}\\}.$ (45)

The second type is defined by

$\phi_{j}\to\phi_{j}^{\prime}=\phi_{j}\pm
i\sqrt{\varepsilon}g_{j\ell}\phi_{\ell},$ (46)
$\phi_{k}^{*}\to\phi_{k}^{\prime*}=\phi_{k}^{*}\mp
i\sqrt{\varepsilon}g_{km}^{*}\phi_{m}^{*},$ (47)

and occurs with transition rate $\alpha_{\ell m,jk}\geq 0$, which holds the
property

$\alpha_{\ell m,jk}=\alpha_{m\ell,kj}.$ (48)

The contribution of this transition to the right-hand side of equation (42) is
given by the expression

$2\sum_{jk\ell m}\alpha_{\ell m,jk}\frac{\partial^{2}{\cal
Q}}{\partial\phi_{j}\partial\phi_{k}^{*}}g_{j\ell}\phi_{\ell}g_{km}^{*}\phi_{m}^{*},$
(49)

which is real due to the property (48).

The third type is defined by

$\phi_{j}\to\phi_{j}^{\prime}=\phi_{j}-\frac{1}{2}\varepsilon g^{*}_{\ell
j}g_{\ell k}\phi_{k},$ (50)
$\phi_{k}^{*}\to\phi_{k}^{\prime*}=\phi_{k}^{*}-\frac{1}{2}\varepsilon
g^{*}_{\ell j}g_{\ell k}\phi_{j}^{*},$ (51)

occurring with rate $\alpha_{kj,\ell\ell}\geq 0$. The contribution of this
transition to the right-hand side of equation (42) is given by the expression

$-\sum_{\ell}\alpha_{kj,\ell\ell}g^{*}_{\ell j}g_{\ell
k}\\{\frac{\partial{\cal Q}}{\partial\phi_{j}}\phi_{k}+\frac{\partial{\cal
Q}}{\partial\phi_{k}^{*}}\phi_{j}^{*}\\},$

which is real due to the property (48).

Replacing the expressions obtained above in the right-hand side of equation
(42), it becomes identical to the adjoint equation (36), if we set

$\gamma_{km,j\ell}=\alpha_{\ell m,jk}g_{j\ell}g_{km}^{*}.$ (52)

Taking into account that $\gamma_{km,j\ell}$ is the covariance $\langle
G_{km}^{*}G_{j\ell}\rangle$, then the random variables $G_{jk}$ must obey the
relation

$\langle G_{km}^{*}G_{j\ell}\rangle=\alpha_{\ell m,jk}g_{j\ell}g_{km}^{*}.$
(53)

This is accomplished if the absolute value of $G_{jk}$ is a random variable
but not its phase. Indeed, if we write $G_{jk}=R_{jk}e^{ia_{jk}}$ then

$\langle G_{km}^{*}G_{j\ell}\rangle=\langle R_{km}R_{j\ell}\rangle
e^{-ia_{km}}e^{ia_{j\ell}},$ (54)

which is of the form (53).

###  III.3 Central equation

Equation (38) that governs the time evolution of the probability density
distribution $\mathscr{P}(\phi,\phi^{*})$ is the fundamental equation of the
present stochastic approach. From this equation we derive the equation for the
time evolution of any bilinear state function such as th covariances
$\rho_{jk}=\langle\phi_{j}\phi_{k}^{*}\rangle$ of the stochastic variables
$\phi_{j}$. The equation that give the time evolution of $\rho_{jk}$ is
obtained from (38) but we may as well use the equation (36) by replacing
${\cal F}$ by $\phi_{j}\phi_{k}^{*}$. The result is

$\frac{d}{dt}\rho_{jk}=\frac{1}{i\mu}\sum_{\ell}(H_{j\ell}\rho_{\ell
k}-\rho_{j\ell}H_{\ell k})$ $-\sum_{\ell m}(\gamma_{mj,m\ell}\rho_{\ell
k}+\rho_{j\ell}\gamma_{m\ell,mk})+2\sum_{\ell m}\gamma_{km,j\ell}\rho_{\ell
m}.$ (55)

Once this equation is solved, the average $\langle{\cal F}\rangle$ of a
bilinear function ${\cal F}$ is obtained by

$\langle{\cal F}\rangle=\sum_{jk}F_{jk}\rho_{kj}.$ (56)

Equation (55) is the central equation of the present approach and is the most
general form of an equation for the covariance $\rho_{jk}$ that can be derived
from a noise that is linear in $\phi_{j}$ and which conserves in the average
the norm of $\phi_{j}$. The coefficients $\gamma_{\ell j,km}$ are not
arbitrary. As mentioned above, $\gamma_{\ell j,km}^{*}=\gamma_{km,\ell j}$ and
the eigenvalues of the $n^{2}\times n^{2}$ matrix with elements $\gamma_{\ell
j,km}$ are nonnegative.

The central equation (55) can be written in matrix form as follows. We first
define the covariance matrix $\rho$ as the $n\times n$ matrix with elements
$\rho_{jk}$. Then we introduce $n\times n$ matrices $A^{jk}$ whose entries are
all zero except the entry at row $j$ and column $k$ which equals 1. Notice
that $A^{jk}$ is not an entry of a matrix but denotes one of a collection of
$n^{2}$ matrices. Their entries are denoted by $A^{jk}_{\ell m}$ and are given
by

$A^{jk}_{\ell m}=\delta_{j\ell}\delta_{km}.$ (57)

The matrices $A^{jk}$ form a complete basis for the expansion of any $n\times
n$ matrix. For instance, in terms of this set the matrix $\rho$ has the
expansion

$\rho=\sum_{jk}\rho_{jk}A^{jk}.$ (58)

In terms of the basis matrices, the central equation (55) becomes

$\frac{d\rho}{dt}=\frac{1}{i\mu}[H,\rho]$ $+\sum_{jk\ell m}\\!\gamma_{jk,\ell
m}(2A^{\ell m}\rho A^{jk\dagger}-A^{jk\dagger}A^{\ell m}\rho-\rho
A^{jk\dagger}A^{\ell m}),$ (59)

where $[H,\rho]=H\rho-\rho H$ stands for the commutation between the matrices
$H$ and $\rho$.

Using (56), the average of $\langle{\cal F}\rangle$ of a bilinear state
function ${\cal F}$ is determined from $\rho$ by

$\langle{\cal F}\rangle={\rm Tr}F\rho.$ (60)

The matrix $\rho$ holds the following properties. It is a Hermitian matrix
with unit trace,

${\rm Tr}\rho=1,$ (61)

and is a semi-positive definite matrix, which means that its eigenvalues are
non-negative. These properties, which allow us to call $\rho$ a density
matrix, follow from the definition of $\rho$ as a covariance matrix, that is,

$\rho_{jk}=\int\phi_{j}\phi_{k}^{*}\mathscr{P}d\phi d\phi^{*},$ (62)

and from the properties of the distribution density which are $\mathscr{P}\geq
0$ and normalization,

$\int\mathscr{P}d\phi d\phi^{*}=1.$ (63)

The Hermitian property $\rho_{jk}^{*}=\rho_{kj}$ follows from (62). As $\rho$
is Hermitian its eigenvalues are real. To show that the eigenvalues are
nonnegative it suffices to consider a transformation that diagonalizes $\rho$
and use this transformation to change the variables $\phi_{j}$ to new
variables $\phi_{j}^{\prime}$. Taking into account that
$\rho_{jk}=\langle\phi_{j}\phi_{k}^{*}\rangle$ then
$\langle\phi_{j}^{\prime}\phi_{k}^{\prime*}\rangle$ will be diagonal and
coincides with the eigenvalues $p_{j}$ of $\rho$. Therefore,
$p_{j}=\langle\phi_{j}^{\prime}\phi_{j}^{\prime*}\rangle\geq 0$.

The fundamental equation was construct in such a way that the norm was
conserved in the average, which means that

$\langle{\cal N}\rangle=\sum_{j}\rho_{jj}$ (64)

is constant. Choosing this constant to be equal to unity, then ${\rm
Tr}\rho=1$. Alternatively, it follows from equation (59) that $d\,{\rm
Tr}\rho/dt=0$. The semi-definite property of $\rho$ is also preserved at all
times because at any time $\rho_{jk}$ keeps being a covariance.

The above properties are valid as long as $\mathscr{P}$ conserves the
properties of a probability density distribution stated above. But this is
indeed the case as we have demonstrated above.

The equation (59) is formally identical to the quantum master equation
breuer2002 ; manzano2020  introduced by Lindblad lindblad1976  and by Gorini,
Kossakowski, and Sudarshan gorini1976 , describing the time evolution of a
density matrix of a quantum open system. The distinguish feature between them
is that equation (59) was obtained by considering that $\rho$ is a covariance
matrix. Of course, as we have shown above, it turned out to be a density
matrix.

It is worth considering the time evolution of the average
$\chi_{j}=\langle\phi_{j}\rangle$. From (36) we find

$\frac{d\chi_{j}}{dt}=\frac{1}{i\mu}\sum_{k}H_{jk}\chi_{k}-\frac{1}{2}\sum_{k}\gamma_{jk}\chi_{k},$
(65)

where $\gamma_{jk}=\langle K_{jk}\rangle$.

##  IV Quantum mechanics

The noise defined by the last two terms of (24) changes the variable
$\phi_{j}$ by changing both the phase $\theta_{j}$ and the absolute value
$r_{j}$ of $\phi_{j}=r_{j}e^{i\theta_{j}}$. However, if the noise is of the
type given by the equation

$\Delta\phi_{j}=(i\sqrt{\tau}\xi-\frac{\tau}{2}\gamma)\phi_{j},$ (66)

where $\xi$ is a random variable with zero mean and variance $\gamma$, then
the noise still changes the phase $\theta_{j}$ but not the absolute value of
$\phi_{j}$. To show this result it suffices to write this equation in the
equivalent form

$\phi_{j}^{\prime}=e^{i\sqrt{\tau}\gamma\xi}\phi_{j},$ (67)

where $\phi_{j}^{\prime}=\phi_{j}+\Delta\phi_{j}$. This result shows that
$\phi_{j}^{*}\phi_{j}=r_{j}^{2}$ is invariant and the norm ${\cal N}$ is
conserved in the strict sense.

The noise defined by the expression (66) corresponds to set
$2\gamma_{j\ell,km}=\gamma\delta_{j\ell}\delta_{km}$. The equation (36)
reduces to

$\frac{d}{dt}\langle{\cal F}\rangle=\frac{1}{i\mu}\langle\\{{\cal F},{\cal
H}\\}\rangle$ $-\frac{1}{2}\gamma\sum_{j}\langle\phi_{j}\frac{\partial{\cal
F}}{\partial\phi_{j}}+\phi_{j}^{*}\frac{\partial{\cal
F}}{\partial\phi_{j}^{*}}\rangle+\gamma\sum_{jk}\langle\phi_{j}\phi_{k}^{*}\frac{\partial^{2}{\cal
F}}{\partial\phi_{j}\partial\phi_{k}^{*}}\rangle,$ (68)

and the fundamental equation (38) becomes

$\frac{d\mathscr{P}}{dt}=\frac{1}{i\mu}\\{{\cal H},\mathscr{P}\\}$
$+\frac{\gamma}{2}\sum_{j}(\frac{\partial\phi_{j}\mathscr{P}}{\partial\phi_{j}}+\frac{\partial\phi_{j}^{*}\mathscr{P}}{\partial\phi_{j}^{*}})+\gamma\sum_{jk}\frac{\partial^{2}\phi_{j}\phi_{k}^{*}\mathscr{P}}{\partial\phi_{j}\partial\phi_{k}^{*}},$
(69)

which reduces to the simpler form

$\frac{\partial\mathscr{P}}{\partial t}=\frac{1}{i\mu}\\{\mathscr{P},{\cal
H}\\}+\gamma^{2}\sum_{jk}\frac{\partial^{2}\mathscr{P}}{\partial\theta_{j}\partial\theta_{k}}.$
(70)

In this case the equation (55) becomes

$\frac{d}{dt}\rho_{jk}=\frac{1}{i\mu}\sum_{\ell}(H_{j\ell}\rho_{\ell
k}-\rho_{j\ell}H_{\ell k}),$ (71)

which can be written in the matrix form as

$\frac{d\rho}{dt}=\frac{1}{i\mu}[H,\rho],$ (72)

which is the quantum Liouville equation, if we set $\mu=\hbar$. The equation
(65) for $\chi_{j}$ becomes

$\frac{d\chi_{j}}{dt}=\frac{1}{i\mu}\sum_{k}H_{jk}\chi_{k}-\frac{1}{2}\gamma\chi_{j}.$
(73)

We remark that the equation (71) does not have the terms corresponding to the
noise. However, this does not mean that the effect of the noise is not
present. The variable $\phi_{j}$ is still a stochastic variable which is
reflectedin the last term of equation (73).

The equation (71) has a special type of solution which is
$\rho_{jk}=\psi_{j}\psi_{k}^{*}$. Replacing this form of $\rho$ in the
equation (71), we see that it is indeed a solution as long as $\psi_{j}$
fulfills the equation

$\frac{d\psi_{j}}{dt}=\frac{1}{i\mu}\sum_{k}H_{jk}\psi_{k},$ (74)

or in matrix form

$\frac{d\psi}{dt}=\frac{1}{i\mu}H\psi,$ (75)

where $\psi$ is a column matrix with elements $\psi_{j}$, and

$\rho=\psi\psi^{\dagger},$ (76)

were $\psi^{\dagger}$ is a row matrix with elements $\psi_{j}^{*}$. A solution
of this type is called pure state and equation (75) is identified as the
Schrödinger equation, if we set $\mu=\hbar$. We remark that $\psi_{j}$ should
not be confused with $\phi_{j}$ nor with $\chi_{j}=\langle\phi_{j}\rangle$.
This last quantity obeys the equation (73) which differs from (74) by the
presence of the term $-\gamma\chi_{j}/2$ which makes $\chi_{j}$ to vanish in
the long term.

The relation between $\phi_{j}$ and $\psi_{j}$ is

$\psi_{j}\psi_{k}^{*}=\langle\phi_{j}\phi_{k}^{*}\rangle=\rho_{jk},$ (77)

and in particular

$|\psi_{j}|^{2}=\langle|\phi_{j}|^{2}\rangle=\rho_{jj}.$ (78)

Due to the invariance of the norm given by (64) it follows the usual
normalization of $\psi_{j}$,

$\sum_{j}|\psi_{j}|^{2}=1.$ (79)

The average $\langle{\cal F}\rangle$ given by equation (56) now reads

$\langle{\cal F}\rangle=\sum_{jk}\psi_{k}^{*}F_{jk}\psi_{j},$ (80)

which is the usual expression for the quantum average.

##  V Quantum Thermodynamics

###  V.1 First law of thermodynamics

We start by writing the central equation (59) in a more convenient form as
follows

$\frac{d\rho}{dt}=\frac{1}{i\mu}[H,\rho]+L,\qquad L=\sum_{r}L^{r}$ (81)

where

$L^{r}=\frac{1}{2}\sum_{s}\\{\gamma_{r,s}[A^{s}\rho,A^{r\dagger}]+\gamma_{r,s}^{\prime}[A^{s}\rho,A^{r}]$
$+\gamma_{r,s}^{*}[A^{r},\rho
A^{s\dagger}]+\gamma_{r,s}^{\prime*}[A^{r\dagger},\rho A^{s\dagger}]\\},$ (82)

and we are using a convention concerning the index notation defined by the
replacements $jk\to r$ and $\ell m\to s$, and $\gamma_{jk,\ell
m}^{\prime}=\gamma_{kj,\ell m}$, and we remark that $L^{r\dagger}=L^{r}$.
Defining the auxiliary matrices

$B^{r}=\sum_{s}\gamma_{rs}A^{s},\qquad
C^{r}=\sum_{s}\gamma_{r,s}^{\prime}A^{s},$ (83)

then $L^{r}$ acquires the simpler form

$L^{r}=\frac{1}{2}\\{[A^{r},\rho
B^{r\dagger}-C^{r}\rho]-[A^{r\dagger},B^{r}\rho-\rho C^{r\dagger}]\\}.$ (84)

Notice that $B^{jk}_{\ell m}=\gamma_{jk,\ell m}$ and $C^{jk}_{\ell
m}=\gamma_{kj,\ell m}$.

From the central equation (81) we determine the equation that gives the time
evolution of the average $\langle{\cal F}\rangle={\rm Tr}(F\rho)$ of a
bilinear state function. It is given by

$\frac{d\langle{\cal F}\rangle}{dt}=\frac{1}{i\mu}{\rm
Tr}F[H,\rho]+\sum_{r}{\rm Tr}FL^{r}.$ (85)

In accordance with thermodynamics the change in the energy ${\cal E}$ of an
open system equals the heat introduced into the system minus the work done by
the system on the environment, which is the law of conservation of energy, or
the first law of thermodynamics. The work is assumed to be the increment in a
potential ${\cal V}$ due to the external forces. The conservation of energy is
then written as

$\frac{d\langle{\cal E}\rangle}{dt}=\Phi-\frac{d\langle{\cal V}\rangle}{dt},$
(86)

where $\Phi$ is the flux of heat into the system. We assume that the
Hamiltonian function ${\cal H}$ is the sum of the energy function ${\cal E}$
and the potential ${\cal V}$ of the external forces, ${\cal H}={\cal E}+{\cal
V}$. Bearing this in mind, we write

$\frac{dU}{dt}=\Phi,$ (87)

where $U=\langle{\cal H}\rangle$.

To determine the expression of $\Phi$, we calculate the left-hand side of
equation (87) using equation (85). The result for the flux is

$\Phi=\sum_{r}\Phi_{r},$ (88)

where

$\Phi_{r}={\rm Tr}HL^{r}.$ (89)

Equation (87), with $\Phi$ representing the total flux of heat into the
system, expresses the conservation of energy, or the first law of
thermodynamics.

###  V.2 Second law of thermodynamics

If we wish to describe an open system regarded as a thermodynamic system in
the sense that it obeys the laws of thermodynamics we need to introduce two
quantities which are the entropy $S$ of the system and the flux of entropy
$\Psi$.

Let $f(x)\leq 0$ be a decreasing and convex function of $x$ defined on the
interval $0\leq x\leq 1$, and such that $f(1)=0$. We also require that
$xf(x)\to 0$ when $x\to 0$. An example of $f(x)$ holding these properties is
the function $-\ln x$. The entropy $S$ of the system is then defined by

$S=\kappa\sum_{j}p_{j}f(p_{j}),$ (90)

where $\kappa$ is some positive constant and $p_{j}\geq 0$ are the eigenvalues
of the matrix $\rho$, which are subject to the condition

$\sum_{j}p_{j}=1,$ (91)

following from ${\rm Tr}\rho=1$.

Considering that $0\leq p_{j}\leq 1$ it follows that $S\geq 0$. The minimum
value of the entropy is $S=0$ which occurs when one of the quantities $p_{j}$
equals one, which is the case of pure states. In terms of the matrix $\rho$,
the entropy can be written as

$S=\kappa\,{\rm Tr}\rho f(\rho).$ (92)

Deriving $S$ with respect to time and using the central equation (81), we find

$\frac{dS}{dt}=\kappa\sum_{r}{\rm Tr}L^{r}f(\rho).$ (93)

By calling $S$ the entropy we wish that it describes the thermodynamic
entropy. A distinguishing property of thermodynamic entropy is that it is not
a conserved quantity. For instance, a system that has its energy increased
only because of heat flow must have its entropy increased. Therefore, the
right-hand side of (93) is not in general the entropy flux.

Concerning the entropy flux, we assume with Clausius that the entropy flux is
the ratio between the heat flux and the temperature of the environment. More
precisely, we assume that the total entropy flux is a sum of entropy fluxes
coming from distinct parts of the environment which are at distinct
temperatures. The entropy flux $\Psi_{r}$ coming from the $r$-th part of the
environment at temperature $T_{r}$ is

$\Psi_{r}=\frac{\Phi_{r}}{T_{r}},$ (94)

where $\Phi_{r}$ is the heat flux coming from the $r$-th part which we assume
to be given by the expression (89). However, we cannot adopt the Clausius
relation in the strict sense because no temperature has yet been defined. But
we can still assume the Clausius relation, with the understanding that
$T_{r}>0$ are parameters of the present approach.

Replacing the expression for the heat flux given by (89) in (94), we find the
desired expression for the flux of entropy coming from the $r$-th part of the
environment,

$\Psi_{r}=\frac{1}{T_{r}}{\rm Tr}HL^{r}.$ (95)

The total flux of entropy is

$\Psi=\sum_{r}\Psi_{r}.$ (96)

The right-hand side of (93) must be the sum of the entropy flux $\Psi$ from
the environment and another term corresponding to the creation of entropy
which is the rate of entropy production $\Pi$ within the system, defined by

$\frac{dS}{dt}=\Pi+\Psi,$ (97)

From this relation we determine $\Pi$ using the expressions of the flux of
entropy $\Psi$ and $dS/dt$. Before that, we define $\rho^{r}$ through the
relation

$\beta_{r}H=f(\rho^{r})+c_{r},$ (98)

where $c_{r}$ is such that ${\rm Tr}\rho_{r}=1$, and $\beta_{r}=1/\kappa
T_{r}$. From this relation and equation (95) we may write $\Psi_{r}$ as

$\Psi_{r}=\kappa{\rm Tr}L^{r}f(\rho^{r}),$ (99)

where the constant $C_{r}$ has disappeared because ${\rm Tr}L^{r}=0$.
Subtracting (99) from (93), we find

$\Pi=\kappa\sum_{r}{\rm Tr}L^{r}\\{f(\rho)-f(\rho^{r})\\}.$ (100)

According to Clausius the increase in the entropy $S$ of a system is larger
than or equal to the entropy flux $\Psi$ into the system,

$\frac{dS}{dt}\geq\Psi,$ (101)

which constitutes the second law of thermodynamics. Taking into account the
equality (97), the Clausius expression (101) of the second law can be written
in the equivalent form

$\Pi\geq 0.$ (102)

In the following, we will demonstrate this inequality proving thus the
Clausius expression for the second law of thermodynamics.

###  V.3 Positivity of the entropy production

To show that $\Pi\geq 0$, we demonstrate that each term of the summation in
(100) is nonnegative, that is, we demonstrate that $\Pi_{r}\geq 0$,

$\Pi_{r}=\kappa{\rm Tr}L^{r}(\rho)\\{f(\rho)-f(\rho^{r})\\},$ (103)

and $L^{r}$ is given by (84).

Our procedure is to show (a) that $\Pi_{r}$ is a convex function of the
eigenvalues $p_{\ell}$ of $\rho$, and (b) that $\Pi_{r}$ is bounded from below
at $\rho=\rho^{r}$, in which case $\Pi_{r}=0$. If the first proposition is
demonstrated, then the second proposition can be met if $\rho=\rho^{r}$ is a
double zero of $\Pi_{r}$ which amounts to say that $L^{r}$ vanishes when
$\rho=\rho^{r}$. This condition leads consistently to the vanishing of
$\Psi_{r}$ as well. The vanishing of $L^{r}$ is fulfilled by demanding that

$B^{r}\rho^{r}=\rho^{r}C^{r\dagger},$ (104)

which we assume from now on. As $B^{r}$ and $C^{r}$ are defined in terms of
$\gamma_{r,s}$ and $\rho^{r}$ depends on $T_{r}$, this condition establishes a
relation between the correlations $\gamma_{r,s}$ and the parameter $T_{r}$.

To show that $\Pi_{r}$ is convex it suffices to show that

$R_{r}={\rm Tr}L^{r}(\rho)f(\rho)$ (105)

is convex because $L^{r}(\rho)$ is linear in $\rho$. We provide the
demonstration of convexity of $R_{r}$ for the case in which $L(\rho)$ has a
diagonal form, which is equivalent to express $L^{r}(\rho)$ as

$L^{r}=\frac{\lambda_{r}}{2}\\{[A^{r},[\rho,A^{r\dagger}]]+[A^{r\dagger},[\rho,A^{r}]]\\},$
(106)

in which case

$R_{r}=\lambda_{r}\Re{\rm
Tr}\\{[f,A^{r\dagger}\rho]A^{r}+[f,A^{r}\rho]A^{r\dagger}\\}.$ (107)

Next, we consider the following term of $R_{r}$,

${\rm Tr}[f,A\rho]A^{\dagger},$ (108)

where for simplicity we have dropped the index of $A^{r}$, and we observe that
it is the limit when $\varepsilon\to 0$ of the expression

$\frac{1}{\varepsilon}{\rm Tr}\\{e^{\varepsilon f}A\rho\,e^{-\varepsilon
f}A^{\dagger}-A\rho A^{\dagger}\\}.$ (109)

The expression (108) is convex if the first term of (109) is convex as the
second term is linear in $\rho$. To see that this is the case it suffices to
write it in the following symmetric form

${\rm Tr}\\{(e^{\varepsilon f/2}A\rho^{1/2}e^{-\varepsilon
f/2})(e^{-\varepsilon f/2}\rho^{1/2}A^{\dagger}e^{\varepsilon f/2}\\}.$ (110)

In the general case, instead of terms of the type (108), we face terms of the
type

${\rm Tr}[f,A^{r}\rho]B^{r\dagger},$ (111)

which is equal to

$\sum_{s}\gamma_{rs}^{*}{\rm Tr}[f,A^{r}\rho]A^{s\dagger}.$ (112)

We then expand $A^{r}$ in another basis set $A^{\prime s}$ in such a way that
the expression (112) becomes diagonal in $A^{\prime s}$, that is,

$\sum_{s}\lambda_{s}{\rm Tr}[f,A^{\prime s}\rho]A^{\prime s\dagger}.$ (113)

Now each of the terms of the summation is similar to (108) and we may apply
the same reasoning above to show that it is convex. This concludes the
demonstration that $\Pi_{r}\geq 0$ and $\Pi\geq 0$.

##  VI Fokker-Planck-Kramers structure

###  VI.1 Fluctuation and dissipation

The classical Fokker-Planck-Kramers (FPK) equation describes a massive
particle under the action of a conservative force plus, a dissipative force,
and a fluctuating force vankampen1981 ; tome2015 . These forces give rise to
three terms in the equation which are the Hamiltonian part, the dissipative
part, and the fluctuation part. In the following we show that the central
equation has a similar structure and can thus be understood as the quantum
version of the FPK equation.

From the expressions (81) and (84), the central equation can be written in the
form

${i\mu}\frac{d\rho}{dt}=[H,\rho]-\sum_{r}[A^{r},J^{r\dagger}],$ (114)

where $J^{r}$ is the current, given by

$J^{r}=i\mu(B^{r}\rho-\rho C^{r\dagger}).$ (115)

Defining $D^{r}=B^{r\dagger}-C^{r}$, the current becomes

$J^{r}=i\mu(D^{r\dagger}\rho-[\rho,C^{r\dagger}]).$ (116)

Replacing it in (114), the central equation, it acquires the form

${i\mu}\frac{d\rho}{dt}=[H,\rho]+i\mu\sum_{r}[A^{r},\rho
D^{r\dagger}]-i\mu\sum_{r}[A^{r},[C^{r},\rho]],$ (117)

which has the form of the FPK equation. The three terms on the right-hand side
are respectively, the Hamiltonian part, the dissipative part, and the
fluctuation part.

Using the relation (104), we find the following form for the term related to
the dissipation,

$D^{r}=(\rho^{r})^{-1}C^{r}\rho^{r}-C^{r}.$ (118)

Therefore, the equation (114) can be set up if we are given $C^{r}$ and
$\rho^{r}$.

###  VI.2 Detailed balance

The stationary solution $\rho^{\rm st}$ of the central equation (114) is
determined by

$[H,\rho^{\rm st}]-\sum_{r}[A^{r},J^{r\dagger}(\rho^{\rm st})]=0.$ (119)

Depending on the covariances of the independent random variables, that is, on
$B^{r}$ and $C^{r}$, the stationary solution may in addition obey a detailed
balance condition. Denoting in this case the stationary solution by $\rho^{\rm
e}$, the detailed balance condition is equivalent to say that each term of
(119) vanishes, that is, $[H,\rho^{\rm e}]=0$ and $J^{r\dagger}(\rho^{\rm
e})=0$, or

$B^{r}\rho^{\rm e}=\rho^{\rm e}C^{r\dagger}.$ (120)

for all $r$, which is the expression of the detailed balance condition. This
solution is understood as describing the thermodynamic equilibrium, understood
as the state devoid of currents.

We remark that the detailed balance condition (120) should not be confused
with the relation (104) between $B^{r}$ and $C^{r\dagger}$. If we use relation
(104), the detailed balance condition is just $\rho^{\rm e}=\rho^{r}$, for all
$r$. Bearing in mind that the expression of $\rho^{r}$ for distinct $r$ has
the same form given by equation (98), differing only on the parameter $T_{r}$,
we see that the detailed balance condition occurs if all the parameters
$T_{r}=T$ are equal. In this case we may identify the common value $T$ as the
temperature of the system and the $\rho^{\rm e}$ is given by $\beta
H=f(\rho^{\rm e})+c$, or

$\rho^{\rm e}=f^{-1}(\beta H-c),$ (121)

where $f^{-1}$ is the function inverse of $f$. We recall that the function $f$
is related to the entropy by $S=\kappa{\rm Tr}f(\rho)$.

If $f=-\ln x$, in which case the entropy is given by $S=-\kappa{\rm
Tr}\rho\ln\rho$, then $f^{-1}=e^{-x}$ and

$\rho^{\rm e}=\frac{1}{Z}e^{-\beta H},$ (122)

where $\beta=1/\kappa T$, which is the Gibbs equilibrium state for a system at
a temperature $T$.

###  VI.3 Non-equilibrium stationary states

Writing $L^{r}$ in terms of the current $J^{r}$, the expression for the heat
flux $\Phi_{r}$, the entropy flux $\Psi_{r}=\Phi_{r}/T_{r}$, and the rate of
entropy $\Pi_{r}$ are given by

$\Phi_{r}=\frac{1}{i\mu}{\rm Tr}J^{r\dagger}[A^{r},H],$ (123)
$\Psi_{r}=\frac{\kappa}{i\mu}{\rm Tr}J^{r\dagger}[A^{r},f(\rho^{r})],$ (124)
$\Pi_{r}=\frac{\kappa}{i\mu}{\rm Tr}J^{r\dagger}[A^{r},f(\rho)-f(\rho^{r})].$
(125)

The total heat flux, the total entropy flux and the total entropy rate are are

$\Phi=\sum_{r}\Phi_{r},$ (126)
$\Psi=\sum_{r}\Psi_{r}=\sum_{r}\frac{1}{T_{r}}\Phi_{r},$ (127)
$\Pi=\sum_{r}\Pi_{r}.$ (128)

In the stationary state $\Phi$ and $\Pi+\Psi$ vanish. If in addition the
thermodynamic equilibrium is established, which occurs if all $T_{r}$ are
equal, then all currents $J^{r}$ vanish. In this case $\Phi_{r}$, $\Psi_{r}$,
and $\Pi_{r}$ as well as $\Pi$ and $\Psi$ also vanish. If the stationary state
is not an equilibrium state, which occurs if at least of $T_{r}$ is distinct
from the others, then at least one of the currents $J^{r}$ are nonzero. In
this case, $\Pi$ and $\Psi$ are nonzero, although their sum vanishes. As
$\Pi\geq 0$, then in the nonequilibrium stationary state

$\Pi=-\Psi>0.$ (129)

###  VI.4 Examples

We have proposed a quantum FPK equations which was obtained through the
canonical quantization oliveira2016 . For a system of quantum particles of
mass $m$ the quantum FPK equation reads oliveira2016

$i\hbar\frac{\partial\rho}{\partial
t}=[H,\rho]+\frac{1}{2}\sum_{j}\gamma_{j}[x_{j},\rho
g_{j}+g_{j}^{\dagger}\rho]$
$+\frac{\gamma_{j}m}{i\hbar\beta_{j}}\sum_{j}[x_{j},[x_{j},\rho]],$ (130)

where $x_{j}$ represents the position of particle $j$, and

$g_{j}=-\frac{m}{i\hbar\beta_{j}}(e^{\beta_{j}H}x_{j}e^{-\beta_{j}H}-x_{j}).$

We see that this equation has the form of equation (104) if we set
$\mu=\hbar$, $A_{j}=x_{j}$,

$C_{j}=\frac{\gamma_{j}m}{\hbar^{2}\beta_{j}}x_{j},\quad{\rm and}\quad
D_{j}=\frac{\gamma_{j}g_{j}}{2i\hbar}.$ (131)

The expansion of $g_{j}$ gives

$g_{j}=p_{j}+\frac{\beta_{j}}{2!}[H,p_{j}]+\frac{\beta_{j}^{2}}{3!}[H,[H,p_{j}]]+\ldots$
(132)

where $p_{j}$ is the momentum of particle $j$. As $g_{j}$ is related to
dissipation we see that the quantum friction is not generally proportional to
the momentum as is the case of the classical case, as is manifest in the
classical FKP equation.

For bosons the canonical quantization leads to the following equation
oliveira2018 ; oliveira2023

$i\hbar\frac{\partial\rho}{\partial
t}=[H,\rho]+i\gamma_{j}\sum_{j}\\{[a_{j},\rho
g_{j}^{\dagger}]-[a_{j}^{\dagger},g_{j}\rho]\\}$
$-\frac{i\gamma_{j}}{\beta_{j}}\sum_{j}\\{[a_{j},[a_{j}^{\dagger},\rho]]+[a_{j}^{\dagger},[a_{j},\rho]]\\},$
(133)

where $a_{j}$ and $a_{j}^{\dagger}$ represent, respectively, the annihilation
and creation of a boson in a one-particle state labeled by the index $j$, and

$g_{j}=\frac{1}{\beta_{j}}(e^{-\beta_{j}H_{j}}a_{j}e^{\beta_{j}H_{j}}-a_{j}).$
(134)

Again we see that this equation has the form of equation (104), if we set
$\mu=\hbar$, $A_{j}=a_{j}$,

$C_{j}=\frac{\gamma_{j}}{\hbar\beta_{j}}a_{j},\quad{\rm and}\quad
D_{j}=\frac{\gamma_{j}}{\hbar}g_{j}.$

##  VII Conclusion

We have derived the equations of quantum mechanics and quantum thermodynamics
from the assumption that a quantum system can be described by an underlying
classical system of particles. Each component $\phi_{j}$ of the wave vector
$\phi$is understood as a complex variable whose real and imaginary parts are
proportional to the coordinate and momentum associated to a degree of freedom
of the underlying classical system. The equation of motion is considered to be
a stochastic equation so that $\phi_{j}$ is a stochastic variable. This result
leads us to conclude that the density matrix $\rho$ obeying either the quantum
Liouville or the Lindblad equation is the covariance matrix associated to the
random wave vector $\phi$. In this sense the present approach gives a meaning
to the off-diagonal terms of $\rho$.

The understanding of the wave vector $\phi$ as a stochastic variable and
$\rho$ as its covariance matrix allows an interpretation of quantum mechanics
other than the standard interpretation omnes1994 ; auletta2001 ; freire2022 .
As the trajectory in the Hilbert space are stochastic the present approach may
fit the consistent history interpretation of quantum mechanics griffiths2002
since the several trajectories are possible, each one occurring with a certain
probability. The present approach is also in accordance with the standard
interpretation if we bear in mind that ${\rm Tr}\rho=1$, which means that
$\rho_{ii}$ can be interpreted as a probability.

## Acknowledgment

I wish to thank Jacob Barandes for calling my attention to the papers of
Strocchi and Heslot.

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One of the Odd Zeta Values from $\zeta(5)$ to $\zeta(25)$ Is Irrational

One of the Odd Zeta Values
from $\boldsymbol{\zeta(5)}$ to $\boldsymbol{\zeta(25)}$ Is Irrational.
By Elementary Means††This paper is a contribution to the Special Issue on
Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full
collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html

Wadim ZUDILIN W. Zudilin

Department of Mathematics, IMAPP, Radboud University,
PO Box 9010, 6500 GL Nijmegen, The Netherlands<EMAIL_ADDRESS>http://www.math.ru.nl/~wzudilin/

Received January 31, 2018, in final form March 26, 2018; Published online
March 29, 2018

Available proofs of result of the type ‘at least one of the odd zeta values
$\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational’ make use of the saddle-point
method or of linear independence criteria, or both. These two remarkable
techniques are however counted as highly non-elementary, therefore leaving the
partial irrationality result inaccessible to general mathematics audience in
all its glory. Here we modify the original construction of linear forms in odd
zeta values to produce, for the first time, an elementary proof of such a
result — a proof whose technical ingredients are limited to the prime number
theorem and Stirling’s approximation formula for the factorial.

irrationality; zeta value; hypergeometric series 11J72; 11M06; 33C20

##  1 Introduction

Without touching deeply a history of the question (see [3] for an excellent
account of this), we notice that the irrationality of the _zeta values_ —
values of Riemann’s zeta function

$\displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$

at integers $s=2,3,\dots$, is known for even $s$ and also for $s=3$, while
there are only partial results in this direction for odd $s\geq 5$. A starting
point here has been set in the work [1] with further development, particularly
focusing on $\zeta(5)$, in [7] and [10].

We fix an odd integer $s\geq 7$. Our strategy is constructing two sequences of
linear forms $r_{n}$ and $\hat{r}_{n}$ living in the $\mathbb{Q}$-space
$\mathbb{Q}+\mathbb{Q}\zeta(3)+\mathbb{Q}\zeta(5)+\dots+\mathbb{Q}\zeta(s)$,
for which we have a control of the common denominators $\lambda_{n}$ of
rational coefficients and an elementary access to their asymptotic behaviour
as $n\to\infty$; more importantly, the two coefficients of $\zeta(3)$ in these
forms are proportional (with factor $7$), so that $7r_{n}-\hat{r}_{n}$ belongs
to the space $\mathbb{Q}+\mathbb{Q}\zeta(5)+\dots+\mathbb{Q}\zeta(s)$.
Finally, using $7r_{n}-\hat{r}_{n}>0$ and the asymptotics
$\lambda_{n}(7r_{n}-\hat{r}_{n})\to 0$ as $n\to\infty$ of the linear forms

$\displaystyle\lambda_{n}(7r_{n}-\hat{r}_{n})\in\mathbb{Z}+\mathbb{Z}\zeta(5)+\mathbb{Z}\zeta(7)+\dots+\mathbb{Z}\zeta(s)$

when $s=25$, we conclude that it cannot happen that all the quantities
$\zeta(5),\zeta(7),\dots,\allowbreak\zeta(25)$ are rational.

The original idea of using the so-called well-poised hypergeometric series to
construct linear forms in zeta values of a given parity is due to Ball and
Rivoal [1]; our new ingredient here is using simultaneously such series and
their ‘twists by half’, for an appropriate choice of the parameters. More
precisely, our hypergeometric series assume the form

$\displaystyle
r_{n}=\sum_{\nu=1}^{\infty}R_{n}(\nu)\qquad\text{and}\qquad\hat{r}_{n}=\sum_{\nu=1}^{\infty}R_{n}\big{(}\nu-\tfrac{1}{2}\big{)},$
(1)

where the rational-function summand $R_{n}(t)$ is defined as follows:

$\displaystyle R(t)=R_{n}(t)$
$\displaystyle=\frac{n!^{s-5}\prod_{j=1}^{n}(t-j)\cdot\prod_{j=1}^{n}(t+n+j)\cdot
2^{6n}\prod_{j=1}^{3n}\big{(}t-n-\frac{1}{2}+j\big{)}}{\prod_{j=0}^{n}(t+j)^{s}}$
$\displaystyle=\frac{2^{6n}n!^{s-5}\prod_{j=0}^{6n}\big{(}t-n+\frac{1}{2}j\big{)}}{\prod_{j=0}^{n}(t+j)^{s+1}}.$
(2)

The following Sections 2 and 3 discuss, respectively, the arithmetic and
analysis of the forms (1). In Section 4 we use this information to conclude
with the proof of the claimed result and make some relevant comments.

##  2 Arithmetic ingredients

The notation $d_{n}$ will be used for the least common multiple of
$1,2,\dots,n$. Recall that the prime number theorem is equivalent to the
asymptotics

$\displaystyle\lim_{n\to\infty}d_{n}^{1/n}=e.$ (3)

A rational function $S(t)$ of the form

$\displaystyle
S(t)=\frac{P(t)}{(t-t_{1})^{s_{1}}(t-t_{2})^{s_{2}}\dotsb(t-t_{q})^{s_{q}}},$

whose denominator has degree larger than its numerator, possesses a unique
partial-fraction decomposition

$\displaystyle
S(t)=\sum_{j=1}^{q}\sum_{i=1}^{s_{j}}\frac{b_{i,j}}{(t-t_{j})^{i}}.$

The coefficients here can be computed on the basis of explicit formula

$\displaystyle
b_{i,j}=\frac{1}{(s_{j}-i)!}\bigl{(}S(t)(t-t_{j})^{s_{j}}\bigr{)}^{(s_{j}-i)}\Big{|}_{t=t_{j}}$

for all $i$, $j$ in question. This procedure can be illustrated on the
following examples, in which all the exponents $s_{j}$ are equal to 1:

$\displaystyle\frac{n!}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{(-1)^{k}\binom{n}{k}}{t+k},$
$\displaystyle\frac{\prod_{j=1}^{n}(t-j)}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{(-1)^{n+k}\binom{n+k}{n}\binom{n}{k}}{t+k},$
$\displaystyle\frac{\prod_{j=1}^{n}(t+n+j)}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{(-1)^{k}\binom{2n-k}{n}\binom{n}{k}}{t+k},$
$\displaystyle\frac{2^{2n}\prod_{j=1}^{n}(t+\frac{1}{2}-j)}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{(-1)^{n+k}\binom{2n+2k}{2n}\binom{2n}{n+k}}{t+k},$
$\displaystyle\frac{2^{2n}\prod_{j=1}^{n}(t-\frac{1}{2}+j)}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{\binom{2k}{k}\binom{2n-2k}{n-k}}{t+k},$
$\displaystyle\frac{2^{2n}\prod_{j=1}^{n}(t+n-\frac{1}{2}+j)}{\prod_{j=0}^{n}(t+j)}=\sum_{k=0}^{n}\frac{(-1)^{k}\binom{4n-2k}{2n}\binom{2n}{k}}{t+k}.$

It also means that the function $R(t)$ in (2) can be written as

$\displaystyle R(t)=\sum_{i=1}^{s}\sum_{k=0}^{n}\frac{a_{i,k}}{(t+k)^{i}}$ (4)

with the recipe to compute the coefficients $a_{i,k}$ in its partial-fraction
decomposition. At the same time, the function $R(t)$ is a product of ‘simpler’
rational functions given above, with all coefficients of their partial
fractions being integral.

######  Lemma 2.1.

Let $k_{1},\dots,k_{q}$ be pairwise distinct numbers from the set
$\\{0,1,\dots,n\\}$ and $s_{1},\dots,s_{q}$ positive integers. Then the
coefficients in the expansion

$\displaystyle\frac{1}{\prod_{j=1}^{q}(t+k_{j})^{s_{j}}}=\sum_{j=1}^{q}\sum_{i=1}^{s_{j}}\frac{b_{i,j}}{(t+k_{j})^{i}}$

satisfy

$\displaystyle d_{n}^{s-i}b_{i,j}\in\mathbb{Z},\qquad\text{where}\quad
i=1,\dots,s_{j}\quad\text{and}\quad j=1,\dots,q,$ (5)

where $s=s_{1}+\dots+s_{q}$.

In particular,

$\displaystyle d_{n}^{s-i}a_{i,k}\in\mathbb{Z},\qquad\text{where}\quad
i=1,\dots,s\quad\text{and}\quad k=0,1,\dots,n,$ (6)

for the coefficients in (4).

######  Proof 2.2.

Denote the rational function in question by $S(t)$. The statement is trivially
true when $q=1$, therefore we assume that $q\geq 2$. In view of the symmetry
of the data, it is sufficient to demonstrate the inclusions (5) for $j=1$.
Differentiating a related product $m$ times, for any $m\geq 0$, we obtain

$\displaystyle\frac{1}{m!}\bigl{(}S(t)(t+k_{1})^{s_{1}}\bigr{)}^{(m)}$
$\displaystyle=\frac{1}{m!}\left(\prod_{j=2}^{q}(t+k_{j})^{-s_{j}}\right)^{(m)}=\sum_{\begin{subarray}{c}\ell_{2},\dots,\ell_{q}\geq
0\\\
\ell_{2}+\dots+\ell_{q}=m\end{subarray}}\prod_{j=2}^{q}\frac{1}{\ell_{j}!}\bigl{(}(t+k_{j})^{-s_{j}}\bigr{)}^{(\ell_{j})}$
$\displaystyle=\sum_{\begin{subarray}{c}\ell_{2},\dots,\ell_{q}\geq 0\\\
\ell_{2}+\dots+\ell_{q}=m\end{subarray}}\prod_{j=2}^{q}(-1)^{\ell_{j}}\binom{s_{j}+\ell_{j}-1}{\ell_{j}}(t+k_{j})^{-(s_{j}+\ell_{j})}.$

This implies that

$\displaystyle b_{i,1}=\sum_{\begin{subarray}{c}\ell_{2},\dots,\ell_{q}\geq
0\\\
\ell_{2}+\dots+\ell_{q}=s_{1}-i\end{subarray}}\prod_{j=2}^{q}(-1)^{\ell_{j}}\binom{s_{j}+\ell_{j}-1}{\ell_{j}}\frac{1}{(k_{j}-k_{1})^{s_{j}+\ell_{j}}}$

for $i=1,\dots,s_{1}$. Using $d_{n}/(k_{j}-k_{1})\in\mathbb{Z}$ for
$j=2,\dots,q$ and $\sum_{j=2}^{q}(s_{j}+\ell_{j})=s-i$ for each individual
summand, we deduce the desired inclusion in (5) for $j=1$, hence for any $j$.

The second claim in the lemma follows from considering $R(t)$ as a product of
the ‘simpler’ rational functions.

######  Lemma 2.3.

For the coefficients $a_{i,k}$ in (4), we have

$\displaystyle a_{i,k}=(-1)^{i-1}a_{i,n-k}\qquad\text{for}\quad
k=0,1,\dots,n\quad\text{and}\quad i=1,\dots,s,$

so that

$\displaystyle\sum_{k=0}^{n}a_{i,k}=0\qquad\text{for}\ i\ \text{even}.$

######  Proof 2.4.

Since $s$ is odd, the function (2) possesses the following (well-poised)
symmetry: $R(-t-n)=-R(t)$. Substitution of the relation into (4) results in

$\displaystyle-\sum_{i=1}^{s}\sum_{k=0}^{n}\frac{a_{i,k}}{(t+k)^{i}}$
$\displaystyle=\sum_{i=1}^{s}\sum_{k=0}^{n}\frac{a_{i,k}}{(-t-n+k)^{i}}=\sum_{i=1}^{s}(-1)^{i}\sum_{k=0}^{n}\frac{a_{i,k}}{(t+n-k)^{i}}$
$\displaystyle=\sum_{i=1}^{s}(-1)^{i}\sum_{k=0}^{n}\frac{a_{i,n-k}}{(t+k)^{i}},$

and the identities in the lemma follow from the uniqueness of decomposition
into partial fractions. The second statement follows from

$\displaystyle\sum_{k=0}^{n}a_{i,k}=(-1)^{i-1}\sum_{k=0}^{n}a_{i,n-k}=(-1)^{i-1}\sum_{k=0}^{n}a_{i,k}.$
∎

######  Lemma 2.5.

For each $n$,

$\displaystyle r_{n}=\sum_{\begin{subarray}{c}i=2\\\
i\;\text{odd}\end{subarray}}^{s}a_{i}\zeta(i)+a_{0}\qquad\text{and}\qquad\hat{r}_{n}=\sum_{\begin{subarray}{c}i=2\\\
i\;\text{odd}\end{subarray}}^{s}a_{i}\big{(}2^{i}-1\big{)}\zeta(i)+\hat{a}_{0},$

with the following inclusions available:

$\displaystyle d_{n}^{s-i}a_{i}\in\mathbb{Z}\qquad\text{for}\quad
i=3,5,\dots,s,\quad\text{and}\quad
d_{n}^{s}a_{0},d_{n}^{s}\hat{a}_{0}\in\mathbb{Z}.$

Notice that

$\displaystyle\big{(}2^{i}-1\big{)}\zeta(i)=\sum_{\ell=1}^{\infty}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}$

for $i\geq 2$.

######  Proof 2.6.

Our strategy here is to write the series in (1) using the partial-fraction
decomposition (4) of $R(t)$. To treat the first sum $r_{n}$ we additionally
introduce an auxiliary parameter $z>0$, which we later specialise to $z=1$:

$\displaystyle r_{n}(z)$
$\displaystyle=\sum_{\nu=1}^{\infty}R_{n}(\nu)z^{\nu}=\sum_{\nu=1}^{\infty}\sum_{i=1}^{s}\sum_{k=0}^{n}\frac{a_{i,k}z^{\nu}}{(\nu+k)^{i}}$
$\displaystyle=\sum_{i=1}^{s}\sum_{k=0}^{n}a_{i,k}z^{-k}\sum_{\nu=1}^{\infty}\frac{z^{\nu+k}}{(\nu+k)^{i}}=\sum_{i=1}^{s}\sum_{k=0}^{n}a_{i,k}z^{-k}\left(\operatorname{Li}_{i}(z)-\sum_{\ell=1}^{k}\frac{z^{\ell}}{\ell^{i}}\right)$
$\displaystyle=\sum_{i=1}^{s}\operatorname{Li}_{i}(z)\sum_{k=0}^{n}a_{i,k}z^{-k}-\sum_{i=1}^{s}\sum_{k=0}^{n}\sum_{\ell=1}^{k}\frac{a_{i,k}z^{-(k-\ell)}}{\ell^{i}},$

where

$\displaystyle\operatorname{Li}_{i}(z)=\sum_{\ell=1}^{\infty}\frac{z^{\ell}}{\ell^{i}}$

for $i=1,\dots,s$ are the polylogarithmic functions. The latter are well
defined at $z=1$ for $i\geq 2$, where $\operatorname{Li}_{i}(1)=\zeta(i)$,
while $\operatorname{Li}_{1}(z)=-\log(1-z)$ does not have a limit as $z\to
1^{-}$. By taking the limit as $z\to 1^{-}$ in the above derivation and using
$R_{n}(\nu)=O\big{(}\nu^{-2}\big{)}$ as $\nu\to\infty$, we conclude that

$\displaystyle\sum_{k=0}^{n}a_{1,k}=\lim_{z\to
1^{-}}\sum_{k=0}^{n}a_{1,k}z^{-k}=0,$

and

$\displaystyle
r_{n}=\sum_{i=2}^{s}\zeta(i)\sum_{k=0}^{n}a_{i,k}-\sum_{i=1}^{s}\sum_{k=0}^{n}a_{i,k}\sum_{\ell=1}^{k}\frac{1}{\ell^{i}}.$
(7)

We proceed similarly for $\hat{r}_{n}$, omitting introduction of the auxiliary
parameter $z$. Since $R(t)$ in (2) vanishes at
$t=-\frac{1}{2},-\frac{3}{2},\dots,-n+\frac{1}{2}$, we can shift the starting
point of summation for $\hat{r}_{n}$ to $t=-m-\frac{1}{2}$, where
$m=\big{\lfloor}\frac{n-1}{2}\big{\rfloor}$, so that

$\displaystyle\hat{r}_{n}$
$\displaystyle=\sum_{\nu=-m}^{\infty}R_{n}\big{(}\nu-\tfrac{1}{2}\big{)}=\sum_{\nu=-m}^{\infty}\sum_{i=1}^{s}\sum_{k=0}^{n}\frac{a_{i,k}}{\big{(}\nu+k-\frac{1}{2}\big{)}^{i}}$
$\displaystyle=\sum_{i=1}^{s}\sum_{k=0}^{n}a_{i,k}\sum_{\nu=-m}^{\infty}\frac{1}{\big{(}\nu+k-\frac{1}{2}\big{)}^{i}}$
$\displaystyle=\sum_{i=1}^{s}\sum_{k=0}^{m}a_{i,k}\sum_{\nu=-m}^{\infty}\frac{1}{\big{(}\nu+k-\frac{1}{2}\big{)}^{i}}+\sum_{i=1}^{s}\sum_{k=m+1}^{n}a_{i,k}\sum_{\nu=-m}^{\infty}\frac{1}{\big{(}\nu+k-\frac{1}{2}\big{)}^{i}}$
$\displaystyle=\sum_{i=1}^{s}\sum_{k=0}^{m}a_{i,k}\left(\sum_{\ell=k-m}^{0}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}+\sum_{\ell=1}^{\infty}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}\right)$
$\displaystyle\quad{}+\sum_{i=1}^{s}\sum_{k=m+1}^{n}a_{i,k}\left(\sum_{\ell=1}^{\infty}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}-\sum_{\ell=1}^{k-m-1}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}\right)$
$\displaystyle=\sum_{i=2}^{s}(2^{i}-1)\zeta(i)\sum_{k=0}^{n}a_{i,k}+\sum_{i=1}^{s}\sum_{k=0}^{m}a_{i,k}\sum_{\ell=0}^{m-k}\frac{(-1)^{i}}{\big{(}\ell+\frac{1}{2}\big{)}^{i}}$
$\displaystyle\quad{}-\sum_{i=1}^{s}\sum_{k=m+1}^{n}a_{i,k}\sum_{\ell=1}^{k-m-1}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}.$
(8)

Now the statement of the lemma follows from the representations in (7) and
(8), Lemma 2.3, the inclusions (6) of Lemma 2.1 and

$\displaystyle
d_{n}^{i}\sum_{\ell=1}^{k}\frac{1}{\ell^{i}}\in\mathbb{Z}\qquad$
$\displaystyle\text{for}\quad 0\leq k\leq n\quad\text{and}\quad i\geq 1,$
$\displaystyle
d_{n}^{i}\sum_{\ell=0}^{m-k}\frac{(-1)^{i}}{(\ell+\frac{1}{2})^{i}}\in\mathbb{Z}\qquad$
$\displaystyle\text{for}\quad 0\leq k\leq m\quad\text{and}\quad i\geq 1,$
$\displaystyle
d_{n-1}^{i}\sum_{\ell=1}^{k-m-1}\frac{1}{\big{(}\ell-\frac{1}{2}\big{)}^{i}}\in\mathbb{Z}\qquad$
$\displaystyle\text{for}\quad m+1\leq k\leq n\quad\text{and}\quad i\geq 1.$ ∎

##  3 Asymptotic behaviour

In this section we make frequent use of Stirling’s asymptotic formula

$\displaystyle n!\sim\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}\qquad\text{as}\quad n\to\infty,$

and its corollary

$\displaystyle\binom{2n}{n}\sim\frac{2^{2n}}{\sqrt{\pi n}}\qquad\text{as}\quad
n\to\infty$

for the central binomial coefficients. (One may also use somewhat weaker but
‘more elementary’ lower and upper bounds

$\displaystyle\int_{1}^{n}\log
x\,{\mathrm{d}}x\leq\log(n!)\leq\int_{2}^{n+1}\log x\,{\mathrm{d}}x$

for the factorial coming out from estimating integral sums of the logarithm
function, with a nemesis of running into more sophisticated versions for the
asymptotics and inequalities below.)

Because the rational function $R_{n}(t)$ in (2) vanishes at $1,2,\dots,n$ and
at $\frac{1}{2},\frac{3}{2},\dots,\allowbreak n-\nobreak\frac{1}{2}$, the
hypergeometric series (1) can be alternatively written as

$\displaystyle
r_{n}=\sum_{\nu=n+1}^{\infty}R_{n}(\nu)=\sum_{k=0}^{\infty}c_{k}\qquad\text{and}\qquad\hat{r}_{n}=\sum_{\nu=n+1}^{\infty}R_{n}(\nu-\tfrac{1}{2})=\sum_{k=0}^{\infty}\hat{c}_{k},$

with the involved summands

$\displaystyle
c_{k}=R_{n}(n+1+k)=\frac{2^{6n}n!^{s-5}\prod_{j=0}^{6n}\big{(}k+1+\frac{1}{2}j\big{)}}{\prod_{j=0}^{n}(n+k+1+j)^{s+1}}=\frac{n!^{s-5}(6n+2k+2)!(n+k)!^{s+1}}{2(2k+1)!(2n+k+1)!^{s+1}}$
(9) and
$\displaystyle\hat{c}_{k}=R_{n}\big{(}n+\tfrac{1}{2}+k\big{)}=\frac{2^{6n}n!^{s-5}\prod_{j=0}^{6n}\big{(}k+\frac{1}{2}+\frac{1}{2}j\big{)}}{\prod_{j=0}^{n}\big{(}n+k+\frac{1}{2}+j\big{)}^{s+1}}$

strictly _positive_. Observe that

$\displaystyle\frac{c_{k}}{\hat{c}_{k}}$
$\displaystyle=\frac{\prod_{j=0}^{6n}(2k+2+j)}{\prod_{j=0}^{6n}(2k+1+j)}\cdot\left(\prod_{j=0}^{n}\frac{n+k+\frac{1}{2}+j}{n+k+1+j}\right)^{s+1}$
$\displaystyle=\frac{6n+2k+2}{2k+1}\cdot\left(2^{-2(n+1)}\frac{\binom{4n+2k+2}{2n+k+1}}{\binom{2n+2k}{n+k}}\right)^{s+1}$
$\displaystyle\sim\frac{6n+2k+2}{2k+1}\left(\frac{n+k}{2n+k+1}\right)^{(s+1)/2}\qquad\text{as}\quad
n+k\to\infty.$ (10)

######  Lemma 3.1.

For $s\geq 7$ odd,

$\displaystyle\lim_{n\to\infty}r_{n}^{1/n}=\lim_{n\to\infty}\hat{r}_{n}^{1/n}=g(x_{0})\qquad\text{and}\qquad\lim_{n\to\infty}\frac{r_{n}}{\hat{r}_{n}}=1$

where

$\displaystyle g(x)=\frac{2^{6}(x+3)^{6}(x+1)^{s+1}}{(x+2)^{2(s+1)}}$

and $x_{0}$ is the unique positive zero of the polynomial

$\displaystyle x(x+2)^{(s+1)/2}-(x+3)(x+1)^{(s+1)/2}.$

######  Proof 3.2.

We have

$\displaystyle\frac{c_{k+1}}{c_{k}}=\frac{\big{(}k+3n+\frac{3}{2}\big{)}(k+3n+2)}{(k+1)\big{(}k+\frac{3}{2}\big{)}}\left(\frac{k+n+1}{k+2n+2}\right)^{s+1}\sim
f\left(\frac{k}{n}\right)^{2}\qquad\text{as}\quad n+k\to\infty,$ (11)

where

$\displaystyle f(x)=\frac{x+3}{x}\left(\frac{x+1}{x+2}\right)^{(s+1)/2}.$

For an ease of notation write $q=(s+1)/2\geq 4$. Since

$\displaystyle\frac{f^{\prime}(x)}{f(x)}=\frac{1}{x+3}-\frac{1}{x}+q\left(\frac{1}{x+1}-\frac{1}{x+2}\right)=\frac{(q-3)x^{2}+3(q-3)x-6}{x(x+1)(x+2)(x+3)}$

and the quadratic polynomial in the latter numerator has a unique positive
zero $x_{1}$, the function $f(x)$ monotone decreases from $+\infty$ to
$f(x_{1})$ when $x$ ranges from $0$ to $x_{1}$ and then monotone increases
from $f(x_{1})$ to $f(+\infty)=1$ (not attaining the value!) when $x$ ranges
from $x_{1}$ to $+\infty$. In particular, there is exactly one positive
solution $x_{0}$ to $f(x)=1$. Notice that $0<x_{0}<1$, because
$f(1)=4\cdot(2/3)^{q}<1$.

The information gained and asymptotics in (11) imply that $c_{k+1}/c_{k}>1$
for the indices $k<x_{0}n-\gamma\sqrt{n}$ and $c_{k+1}/c_{k}<1$ for
$k>x_{0}n+\gamma\sqrt{n}$ for an appropriate choice of $\gamma>0$ dictated by
application of Stirling’s formula to the factorials defining $c_{k}$ in (9)
(see [2, Section 3.4] as well as the second proof of Lemma 3 in [1]). This
means that the asymptotic behaviour of the sum
$r_{n}=\sum_{k=0}^{\infty}c_{k}$ is determined by the asymptotics of
$c_{k_{0}}$ and its neighbours $c_{k}$, where $k_{0}=k_{0}(n)\sim x_{0}n$ and
$|k-k_{0}|\leq\gamma\sqrt{n}$, so that

$\displaystyle\lim_{n\to\infty}r_{n}^{1/n}$
$\displaystyle=\lim_{n\to\infty}c_{k_{0}(n)}^{1/n}$
$\displaystyle=\lim_{n\to\infty}\Biggl{(}\left(\frac{n}{e}\right)^{(s-5)n}\left(\frac{6n+2k_{0}+2}{e}\right)^{6n+2k_{0}+2}\left(\frac{e}{2k_{0}+1}\right)^{2k_{0}+1}$
$\displaystyle\qquad\quad\times\left(\frac{n+k_{0}}{e}\right)^{(s+1)(n+k_{0})}\left(\frac{e}{2n+k_{0}+1}\right)^{(s+1)(2n+k_{0}+1)}\Biggr{)}^{1/n}$
$\displaystyle=\frac{(2x_{0}+6)^{2x_{0}+6}(x_{0}+1)^{(s+1)(x_{0}+1)}}{(2x_{0})^{2x_{0}}(x_{0}+2)^{(s+1)(x_{0}+2)}}$
$\displaystyle=\frac{2^{6}(x_{0}+3)^{6}(x_{0}+1)^{s+1}}{(x_{0}+2)^{2(s+1)}}\cdot
f(x_{0})^{2x_{0}}=g(x_{0}).$

It now follows from (10) that

$\displaystyle\frac{\hat{c}_{k+1}}{\hat{c}_{k}}\sim\frac{c_{k+1}}{c_{k}}\qquad\text{as}\quad
n+k\to\infty,$ (12)

so that the above analysis applies to the sum
$\hat{r}_{n}=\sum_{k=0}^{\infty}\hat{c}_{k}$ as well, and its asymptotic
behaviour is determined by the asymptotics of $\hat{c}_{k_{0}}$ and its
neighbours $\hat{c}_{k}$, where $k_{0}=k_{0}(n)\sim x_{0}n$ and
$|k-k_{0}|\leq\hat{\gamma}\sqrt{n}$. From (12) we deduce that the limits of
$\hat{c}_{k_{0}(n)}^{1/n}$ and $c_{k_{0}(n)}^{1/n}$ as $n\to\infty$ coincide,
hence $\hat{r}_{n}^{1/n}\to g(x_{0})$ as $n\to\infty$. In addition to this, we
also get

$\displaystyle\lim_{n\to\infty}\frac{r_{n}}{\hat{r}_{n}}=\lim_{n\to\infty}\frac{c_{k_{0}(n)}}{\hat{c}_{k_{0}(n)}}=\lim_{n\to\infty}\frac{6n+2k_{0}+2}{2k_{0}+1}\left(\frac{n+k_{0}}{2n+k_{0}+1}\right)^{(s+1)/2}=f(x_{0}),$

which leads to the remaining limiting relation.

##  4 Conclusion

We choose $s=25$ and apply Lemma 3.1 to find out that $7r_{n}-\hat{r}_{n}>0$
for $n$ sufficiently large, and

$\displaystyle\lim_{n\to\infty}(7r_{n}-\hat{r}_{n})^{1/n}=g(x_{0})=\exp(-25.292363\dots),$

where $x_{0}=0.00036713\dots$. Assuming that the odd zeta values from
$\zeta(5)$ to $\zeta(25)$ are all rational and denoting by $a$ their common
denominator, we use Lemma 2.5 and the asymptotics (3) to conclude that the
sequence of _positive integers_

$\displaystyle ad_{n}^{25}(7r_{n}-\hat{r}_{n})$

tends to $0$ as $n\to\infty$; contradiction. Thus, at least one of the numbers
$\zeta(5),\zeta(7),\dots,\allowbreak\zeta(25)$ is irrational.

Those who count the prime number theorem as insufficiently elementary may use
weaker versions of (3), for example, $d_{n}<3^{n}$ from [5] and the choice
$s=33$ instead, to arrive at the same conclusion (for the larger value of $s$,
of course).

Finally, we remark that the novelty of eliminating an ‘unwanted’ term of
$\zeta(3)$ in linear forms in odd zeta values can be further used with the
arithmetic method in [10] to significantly reduce the size of $s$. Since this
does not let $s$ be down to $s=9$, hence leaving the achievement ‘at least one
of the four numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is
irrational’ unchanged, we do not discuss this generalisation in greater
details. We point out, however, that there are other applications of the
hypergeometric ‘twist-by-half’ idea, some discussed in the joint papers [6,
8], and that a far-going extension to general ‘twists’ introduced by J. Sprang
in [9] leads to an elementary proof of a version of the Ball–Rivoal theorem
from [1] as well as to a significant improvement of the latter — see [4] for
details.

### Acknowledgements

I thank Stéphane Fischler, Tanguy Rivoal, Johannes Sprang and the anonymous
referees for their feedback on the manuscript.

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