Appendix F. Time dependences of statistical states
https://doi.org/10.48550/arXiv.2403.13981
A probability distribution function (pdf) for the microstructure or microstate of a classical Hamiltonian system is a state of knowledge. If the deterrministic system it describes is left undisturbed, the pdf evolves smoothly until new information about the physical system is revealed, such as the result of an earlier measurement. New information would change the pdf abruptly in the same way that, when the first lottery number is drawn, the probability that your ticket is a winning ticket abruptly changes.
In classical statistical mechanics, the smooth evolution of the pdf for a system’s microstate is described by the Liouville equation, which is the continuity equation for the flow of probability in phase space expressed in terms of the system’s Hamiltonian [Evans and Morriss, 2008]. If the Hamiltonian does not depend explicitly on time, the flow of probability in phase space is smooth and conserves probability locally in phase space. This means that the probability of the system’s microstate being in any region of phase space can only change via a flow of probability density through the region’s boundaries.
F.1 Evolution of a statistical microstate
Let ${\phasespace\equiv\momspace\times\configspace}$ denote the phase space of a classical system, where $\momspace$ and $\configspace$ are the spaces of all possible momenta and configurations (microstructures), respectively. Let ${\Gammat\equiv(\pi_t,x_t)\in\phasespace}$ denote the system’s true microstate, and its Hamiltonian is \begin{align*}
\smallham:\phasespace\equiv\momspace\times\configspace\to\realone;\; (\pi,x)\mapsto\smallham(\pi,x),
\end{align*}
where ${\pi=(\pi_1,\pi_2,\cdots)}$, ${x=(x_1,x_2,\cdots)}$, and $x_\alpha$ and $\pi_\alpha$ are the coordinate and momentum, respectively, of the ${\alpha^\text{th}}$ degree of freedom.
The partial derivatives with respect to $x_\alpha$ and ${\pi_\alpha}$ are denoted by ${\gradx^\alpha}$ and ${\gradp^\alpha}$, respectively, and we will assume that all second partial derivatives of ${\smallham}$ are continuous. This implies that [Hörmander, 2015]
\begin{align*}
\gradx^\alpha\gradp^\alpha\smallham=\gradp^\alpha\gradx^\alpha\smallham
\implies\gradx\cdot\gradp\smallham=\gradp\cdot\gradx\smallham.
\end{align*}
To define a probability distribution for the location of $\Gammat$ in $\phasespace$, let us use a construction similar to that used in Appendix C.2: Let ${\G\subset\phasespace}$ denote a regular lattice, and let ${\{\N_\Gamma:\Gamma\in\G\}}$ be a partition of $\phasespace$ such that all points in $\phasespace$ that are closer to ${\Gamma\in\G}$ than to any other element of $\G$ are elements of ${\N_\Gamma}$. Since $\G$ is a regular lattice, the measure ${\abs{\N_\Gamma}}$ of ${\N_\Gamma}$ in $\phasespace$ is the same for all ${\Gamma\in\G}$, and is denoted by ${\hilbv>0}$.
Our state of knowledge of $\Gammat$ at time $t$ can be expressed by the probability mass function,
\begin{align*}
\psdist(t)\hilbv:\G\to[0,1];\;\Gamma\mapsto \psdist(\Gamma;t)\hilbv\equiv\Pr(\Gammat\in\N_\Gamma),
\end{align*}
where ${\hilbv}$ is small enough that variations of the probability density function, \begin{align*}
\psdist(t):\phasespace\to\realpos; \;\Gamma\mapsto \psdist(\Gamma;t)=\psdist(\pi,x;t),
\end{align*}
across every subset ${\N_\Gamma}$ in the partition of $\phasespace$ are negligible.
Let us assume that no new information about the physical system is revealed during the interval ${(t,t+\tau)}$, so that ${\psdist(t+\tau)}$ is determined by $\psdist$ evolving smoothly between times $t$ and ${t+\tau}$ as described by the Liouville equation [Evans and Morriss, 2008],
\begin{align*}
\gradt\psdist & + \gradG\cdot\left(\psdist\dot{\Gamma}\right) = 0
\tag{116}
\\
\implies
\gradt\psdist & + \dot{\Gamma}\cdot\gradG\psdist = -\psdist\gradG\cdot\dot{\Gamma} = -\compress\psdist,
\tag{117}
\end{align*}
In Eq. (117), ${\compress=\compress(\Gamma)\equiv \gradG\cdot\dot{\Gamma}(\Gamma)}$ is known as the phase space compression factor [Evans and Morriss, 2008; Evans and Searles, 2002].
If the physical system is isolated, its Hamiltonian $\smallham$ is time-independent. Therefore using Hamilton’s equations, ${\dot{x}_\alpha=\gradp^\alpha\smallham}$ and ${\dot{\pi}_\alpha=-\gradx^\alpha\smallham}$, we find that
\begin{align*}
\compress&=\gradG\cdot\dot{\Gamma}
=\sum_\alpha\left(\gradx^\alpha\dot{x}_\alpha+\gradp^\alpha\dot{\pi}_\alpha\right)
\\
&=\sum_\alpha\left(\gradx^\alpha\gradp^\alpha\smallham-\gradp^\alpha\gradx^\alpha\smallham\right)
=\gradx\cdot\gradp\smallham-\gradp\cdot\gradx\smallham
=0.
\end{align*}
Since the phase space compression factor vanishes, we are left with the simplified Liouville equation, \begin{align*}
\gradt\psdist & + \dot{\Gamma}\cdot\gradG\psdist = 0,
\tag{118}
\end{align*}
which can also be expressed as \begin{align*}
\gradt\psdist &+ \gradx \psdist\cdot\gradp \smallham -\gradp \psdist\cdot\gradx \smallham= 0.
\tag{119}
\end{align*}
Probability must always be conserved, but Eq. (118) and Eq. (119) state that there is local conservation of probability. In other words, the rate of change of the probability density at a point in $\phasespace$ equals the rate at which probability density is flowing into the point from neighbouring points.
If the phase space compression factor were finite, which would be the case if ${\partialt\smallham}$ was finite, an increasing probability of ${\Gammat}$ being in one region of $\phasespace$ could be counterbalanced by a decreasing probability of it being in another region of ${\phasespace}$, with which the first region had no overlap or common boundary points.
There is local conservation of probability if and only if the phase space compression factor vanishes.
F.2 Reduced Liouville equations
Let us express ${\psdist(t)}$ as \begin{align*}
\psdist(\pi,x;t)=\cfgdist(x;t)\momcfgdist(\pi|x;t),
\end{align*}
where ${\momcfgdist(\pi|x;t)}$ is the conditional probability density function for ${\pi_t(t)}$ when it is known that ${x_t(t)=x}$. Then Eq. (119) can be expressed as \begin{align*}
\momcfgdist\gradt \cfgdist &+ \cfgdist\gradt \momcfgdist + \momcfgdist\gradx \cfgdist\cdot\gradp \smallham
\\
&+ \cfgdist\gradx \momcfgdist\cdot\gradp \smallham
-\cfgdist\gradp \momcfgdist\cdot\gradx \smallham = 0.
\end{align*}
If we integrate over the set ${\momspace}$ of accessible momenta, and use the fact that ${\intmom \momcfgdist\dd{\pi}=1}$, which implies that \begin{align*}
\gradt\intmom \momcfgdist\dd{\pi}=\intmom\gradt \momcfgdist \dd{\pi}=0,
\end{align*}
we get \begin{align*}
\gradt \cfgdist + \gradx \cfgdist\cdot&\left(\intmom \dd{\pi} \momcfgdist\gradp \smallham\right)
+ \cfgdist\left(\intmom\dd{\pi}\gradx \momcfgdist\cdot\gradp \smallham\right)
\\
&- \cfgdist\left(\intmom\dd{\pi}\gradp \momcfgdist\cdot \gradx \smallham\right) = 0.
\tag{121}
\end{align*}
The identities \begin{align*}
\intmom\dd{\pi} \gradp\cdot\left(\momcfgdist\partial_x \smallham\right)
&=
\intmom\dd{\pi} \gradp \momcfgdist\cdot \gradx \smallham
\\
&+
\intmom\dd{\pi} \momcfgdist \gradp\cdot\gradx \smallham,
\\
\intmom\dd{\pi}\gradx\cdot\left(\momcfgdist\gradp\smallham\right)
&=
\intmom\dd{\pi}\gradx\momcfgdist\cdot\gradp\smallham
\\
&+\intmom\dd{\pi}\momcfgdist\gradx\cdot\gradp\smallham,
\end{align*}
allow Eq. (121) to be expressed in the form \begin{align*}
\gradt\cfgdist+\gradx\cdot\bigg(\cfgdist\intmom&\dd{\pi}\momcfgdist\gradp\smallham\bigg)
\\
&=\cfgdist\intmom\dd{\pi}\gradp\cdot\left(\momcfgdist\gradx\smallham\right).
\tag{121}
\end{align*}
The generalized Stokes theorem [Baez and Muniain, 1994; Doran and Lasenby, 2007] implies that the integral on the right hand side can be expressed as
\begin{align*}
\intmom\dd{\pi}\gradp\cdot\left(\momcfgdist\gradx\smallham\right)
= \int_{\partial\momspace}\momcfgdist\dd{s_\pi}\left(\hat{s}_\pi\cdot\gradx\smallham\right),
\end{align*}
where ${\partial\momspace}$ is the boundary of set ${\momspace}$; ${\dd{s_\pi}}$ is a volume form of ${\partial\momspace}$; and ${\hat{s}_\pi}$ is a unit vector that is normal to ${\partial\momspace}$ and directed away from set $\momspace$.
By definition of the set ${\momspace}$ of accessible (${\implies}$ possible) momenta, ${\momcfgdist}$ vanishes on ${\partial\momspace}$. Therefore the boundary integral vanishes and Eq. (121) becomes
\begin{align*}
\gradt \cfgdist + \gradx\cdot \left( \cfgdist\intmom\dd{\pi}\momcfgdist\gradp \smallham \right) &= 0,
\tag{122}
\end{align*}
which can also be expressed as \begin{align*}
\gradt \cfgdist + \left(\intmom\dd{\pi}\momcfgdist\gradp \smallham \right)\cdot\gradx\cfgdist &= -\left(\gradx\cdot\intmom\dd{\pi}\momcfgdist\gradp \smallham\right)\cfgdist.
\end{align*}
The quantity in parentheses on the left hand side is the expectation value of ${\dot{x}}$ given that ${x_t=x}$. It will be denoted as \begin{align*}
\barv(x;t)
\equiv
\intmom\dd{\pi}\momcfgdist\gradp \smallham = \intmom\dd{\pi}\momcfgdist(\pi|x;t)\dot{x}(\pi,x).
\tag{123}
\end{align*}
Therefore we can express Eq. (122) in the forms \begin{align*}
\gradt \cfgdist +\gradx\cdot\left( \cfgdist \barv\right) &= 0,
\tag{124}
\\
\gradt\cfgdist + \barv\cdot\gradx\cfgdist &= -\left(\gradx\cdot\barv\right)\cfgdist.
\tag{125}
\end{align*}
Eq. (126) are forms of the reduced Liouville equation in configuration space, $\configspace$.
F.2.1 Reduced Liouville equation in momentum space
If we express $\psdist$ as \begin{align*}
\psdist(\pi,x;t)=\momdist(\pi;t)\cfgmomdist(x|\pi;t),
\end{align*}
a derivation analogous to the derivation of Eq. (126) from Eq. (118) would lead us to \begin{align*}
\gradt \momdist + \gradp\cdot\left(\momdist\barF\right) & =0,
\tag{127}
\\
\gradt\momdist + \barF\cdot\gradp\momdist & =-\left(\gradp\cdot\barF\right)\momdist,
\tag{128}
\end{align*}
where,
\begin{align*}
\barF=\barF(\pi;t)\equiv
\int_\configspace\dd{x} \momdist(x|\pi;t)\dot{\pi}(\pi,x).
\end{align*}
Eq. (129) are forms of the reduced Liouville equation in momentum space, $\momspace$.
F.2.2 Configuration space and momentum space compression factors
Let us define the configuration space compression factor, \begin{align*}
\compressx=\compressx(x;t)\equiv \gradx\cdot\barv(x;t),
\end{align*}
and the momentum space compression factor, \begin{align*}
\compressp=\compressp(\pi;t)\equiv \gradp\cdot\barF(\pi;t).
\end{align*}
Then we can express the reduced Liouville equations (Eq. (125) and Eq. (128)) as \begin{align*}
\gradt \cfgdist +\barv\cdot\gradx\cfgdist &= -\compressx\cfgdist,
\tag{130}
\\
\gradt \momdist +\barF\cdot\gradp\momdist &= -\compressp\momdist.
\tag{131}
\end{align*}
Recall that the flow of probability described by Eq. (117) conserves probability locally if and only if the phase space compression factor ${\compress}$ vanishes everywhere in $\phasespace$. Similarly, the flows of probability described by Eq. (130) and Eq. (131) conserve probability locally if and only if ${\compressx}$ and ${\compressp}$, respectively, vanish everywhere in ${\configspace}$ and ${\momspace}$, respectively.
However $\compressx$ and $\compressp$ are finite, in general, so Eq. (130) and Eq. (131) do not conserve probability locally in $\configspace$ and $\momspace$, respectively. This makes them unsuitable for calculating the simultaneous evolutions of ${\cfgdist}$ and ${\momdist}$, because ${\bar{v}}$ and ${\bar{F}}$ vary in time if $\momcfgdist$ and ${\cfgmomdist}$, respectively, vary in time.
For example, Eq. (130) can only be used to describe the evolution of $\cfgdist$ in a given time interval if ${\barv}$ is known at all times in that interval. However $\momcfgdist$ must be known to calculate $\barv$; and ${\momcfgdist=\psdist/\cfgdist}$ cannot be calculated without knowing ${\psdist}$ and ${\cfgdist}$. Similarly, $\barF$ cannot be known unless $\psdist$ and ${\momdist}$ are known. Therefore Eq. (130) and Eq. (131) could only be used to calculate the simultaneous evolutions of ${\cfgdist}$ and ${\momdist}$ if those evolutions had already been calculated (e.g., from Eq. (118)).
Note that Eqns Eq. (130) and Eq. (131) must conserve total probability. Therefore,
\begin{align*}
\intconfig\gradt\cfgdist(x;t)\dd{x} &= \dvone{t}\left(\intconfig\cfgdist(x;t)\dd{x}\right) = 0
\\
\implies \intconfig\barv(x)\cdot\gradx\cfgdist(x)\dd{x}&= -\intconfig\compressx(x)\cfgdist(x)\dd{x},
\end{align*}
and \begin{align*}
\intmom\gradt\momdist(\pi)\dd{\pi} &= \dvone{t}\left(\intmom\momdist(\pi)\dd{\pi}\right) = 0,
\\
\implies \intmom\barF(\pi)\cdot\gradx\momdist(\pi)\dd{\pi}&= -\intmom\compressp(\pi)\momdist(\pi)\dd{\pi}.
\end{align*}
SUBSECTION MISSING (mid rewrite)
F.3 Evolutions of microstructures in Hilbert spaces
Let ${\hilbert_x}$ denote the Hilbert space whose rays correspond to microstructure pdfs, such as ${\cfgdist(t)}$. Let ${\ket{\psi(t)}}$ and ${\ket{\psi(t+\tau)}}$ be elements of ${\hilbert_x}$ which possess the same information as ${\cfgdist(t)}$ and ${\cfgdist(t+\tau)}$, respectively. That information could be retrieved via Eq. (106) and Eq. (109), as follows: \begin{align*}
\cfgdist(x;t) & = \frac{\braket{\psi(t)}{x}\braket{x}{\psi(t)}}{\braket{\psi(t)}{\psi(t)}}
= \frac{\psi^*(x;t)\psi(x;t)}{\int_\configspace \psi^*(x';t)\psi(x';t)\dd{x'}}.
\end{align*}
Let us choose the set \begin{align*}
\left\{\ket{\psi(t')}:t\leq t'\leq t+\tau\right\}
\end{align*}
of state vectors at times between $t$ and ${t+\tau}$ such that they all have unit norms (${\braket{\psi}{\psi}=1}$) and such that the evolution of ${\ket{\psi(t)}}$ into ${\ket{\psi(t+\tau)}}$ is differentiable with respect to time.
Let us also choose each state ${\ket{\psi(t')}}$ along this state trajectory such that the wavefunction,
\begin{align*}
\psi(x;t')\equiv\braket{x}{\psi(t')}=\abs{\braket{x}{\psi(t')}}e^{2\pi i\theta(x;t')},
\end{align*}
is a differentiable function of $x$. As discussed in Sec. 3.4.2 and Appendix C, this is even possible at a point ${x_0\in\configspace}$ where ${\cfgdist}$ vanishes, but its gradient ${\gradx \cfgdist(x)}$ remains finite as ${x\to x_0}$. In other words, even if $\cfgdist$ has a non-differentiable cusp at $x_0$, ${\psi}$ can be differentiable at $x_0$ as long as the phase factor ${e^{2\pi i\theta(x;t')}}$ changes sign at $x_0$.
Because the state trajectory is smooth and norm-preserving, ${\ket{\psi(t)}}$ and ${\ket{\psi(t+\tau)}}$ are related by a rotation. In other words, for every ${\tau\in\realpos}$, there exists a unitary operator ${
\hU(\tau):\hilbert_x\to\hilbert_x}$, which varies smoothly with $\tau$, such that
\begin{align*}
\ket{\psi(t+\tau)} = \hU(\tau)\ket{\psi(t)}.
\tag{132}
\end{align*}
${\hU(\tau)}$ is independent of $t$ because $\cfgdist$ evolves according to Eq. (126) and its evolution from time $t$ depends only on ${\cfgdist(t)}$ and the system’s Hamiltonian, ${\smallham}$, which is time-independent. Therefore, given any time ${\tilde{t}\neq t}$, if ${\cfgdist(\tilde{t})=\cfgdist(t)}$, then ${\cfgdist(\tilde{t}+\tau)=\cfgdist(t+\tau)}$. This implies that the information shared by ${\ket{\psi(t)}}$, ${p(t)}$, and ${p(\tilde{t})}$ could also be expressed as a unit vector ${\ket{\phi(\tilde{t})}\in \hilbert_x}$, and ${\hU(\tau)\ket{\phi(\tilde{t})}}$ would possess the same physical information as ${\cfgdist(\tilde{t}+\tau)}$.
Another way to say this is that time is homogeneous for the physical system, and for the evolution of what is known about it, because the system is isolated. Its isolation implies that $\smallham$ does not depend explicitly on time, and that the evolution of $\cfgdist$ between time $t$ and time ${t+\tau}$ depends only on ${\cfgdist(t)}$ and on $\smallham$.
Every unitary operator can be expressed as the exponential of an anti-Hermitian operator. Therefore, let
\begin{align*}
\hU(\tau)\equiv e^{-2\pi i\hB(\tau)/h},
\end{align*}
where ${\hB(\tau)}$ is Hermitian, which means that ${-i\hB(\tau)}$ is anti-Hermitian, and $h$ is the constant with dimensions ${\text{energy}\times\text{time}}$ that was introduced in [section:CMQM_boundary]. Smooth evolution of ${\ket{\psi}}$ implies that ${\hB(\tau)/h}$ must vanish in the limit ${\tau\to 0}$. Therefore, in this limit we can express Eq. (132) as \begin{align*}
\ket{\psi(t+\tau)} &= e^{-2\pi i\hB(\tau)/h}\ket{\psi(t)}
\\
& =\left(1-2\pi i\hB(\tau)/h\right)\ket{\psi(t)}+\order{\hB^2/h^2}
\\
\implies \gradt\ket{\psi(t)}&\equiv \lim_{\tau\to 0}\frac{\ket{\psi(t+\tau)}-\ket{\psi(t)}}{\tau}
\\
& = -\frac{i}{\hbar}\left(\lim_{\tau\to 0}\frac{\hB(\tau)}{\tau}\right)\ket{\psi(t)},
\tag{133}
\end{align*}
where ${\hbar\equiv h/(2\pi)}$. We have not made any assumptions about the state ${\ket{\psi(t)}}$ to derive this equation, which means that it is generally valid and that ${\hU(\tau)}$, and hence ${\hB(\tau)}$, are $t$-independent properties of the physical system.
If we define the system-dependent operator,
\begin{align*}
\hH_x\equiv \lim_{\tau\to 0} \left(\frac{\hB(\tau)}{\tau}\right),
\end{align*}
then Eq. (133) becomes \begin{align*}
\hH_x\ket{\psi}=i\hbar\gradt\ket{\psi}.
\tag{134}
\end{align*}
What this shows is that, when a classical statistical microstate, ${\cfgdist(x)}$, is expressed as an element ${\ket{\psi}}$ of a Hilbert space, the equation governing its time-evolution has the same mathematical form as the time-dependent Schrödinger equation. The analogous equation, \begin{align*}
\hH_p\ket{\psi_p}=i \hbar\gradt\ket{\psi_p},
\tag{135}
\end{align*}
could be derived from Eq. (129) to describe the evolution of an element ${\ket{\psi_p}}$ of a Hilbert space $\hilbert_p$, which contains the same physical information as the statistical momentum state, ${\tilde{\momcfgdist}}$.
Note that I have used suggestive notation to make the similarity between Eq. (134) and Eq. (135) and the time-dependent Schrödinger more obvious.
F.4 Schr\"odinger equation (generic form, position representation)
Now let us try to find a more explicit form for ${\hH_x}$, when it is represented in the basis of microstructures, $\ket{x}$, by attempting to recover Eq. (126) from Eq. (134). After projecting both sides of Eq. (134) onto ${\ket{x}}$, and inserting the identity, we get \begin{align*}
\mel{x}{\hH_x\hone}{\psi_x}
&= \bra{x}\hH_x\left(\int_\configspace \dyad{x'}\dd{x'}\right)\ket{\psi_x}
= i\hbar \gradt\braket{x}{\psi_x}
\\
\implies &
\int_\configspace H_x(x,x')\psi_x(x')\dd{x'} = i \hbar \gradt\psi_x(x)
\end{align*}
where ${\psi_x(x)\equiv\braket{x}{\psi_x}}$ and ${H_x(x,x')\equiv\mel{x}{H_x}{x'}}$. Let us use ${\smallhamop}$ to denote the integral operator that acts on $\psi_x$ on the left hand side of this equation, so that we can write it as \begin{align*}
\smallhamop\psi_x = i\hbar \gradt \psi_x.
\end{align*}
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