Finite reversible stochastic rate model and thermal coupling

Assume the Axiom of Choice The Axiom of Choice. A finite reversible stochastic rate model consists of an integer $r\ge 2$, the finite state set $E=\{1,\dots ,r\}$, real energy parameters $e_a\in \mathbb{R}$, an inverse-energy parameter $\beta >0$, and finite transition rates $w_{ab}\ge 0$ for distinct $a,b\in E$. When physical units are assigned, $e_a$ is measured in joules, $\beta$ in joules${}^{-1}$, and $w_{ab}$ in seconds${}^{-1}$; this mathematical definition does not assert a physical realization. Define

$Z:={\sum }_{a\in E}{e}^{-\beta e_a},{\pi }_a:=Z^{-1}{e}^{-\beta e_a}\mathrm{.}$

Then $Z$ is finite and positive, and $\pi$ is a strictly positive probability vector. Require $w_{ab}>0$ exactly when $w_{ba}>0$, require the undirected graph with edge $\{a,b\}$ whenever $w_{ab}>0$ to be connected, and impose detailed balance ${\pi }_a{w}_{ab}={\pi }_b{w}_{ba}$. Define the row generator $Q$ by $Q_{ab}=w_{ab}$ for $a\mathrm{\ne }b$ and $Q_{aa}=-q_a$, where $q_a:={\sum }_{b\mathrm{\ne }a}{w}_{ab}$. Since $r\ge 2$ and the graph is connected, every $q_a>0$. Thus $(Qf{)}_a={\sum }_{b\mathrm{\ne }a}{w}_{ab}(f_b-f_a)$ for a column observable $f$, and a row probability vector evolves by $p^{\mathrm{\prime }}=pQ$.

For any observable $f:E\to \mathbb{R}$, write $\pi (f):={\sum }_{a\in E}{\pi }_a{f}_a$. For $f,g:E\to \mathbb{R}$, define $⟨f,g{⟩}_{\pi }:={\sum }_{a\in E}{\pi }_a{f}_a{g}_a$. Since $E$ is finite and every ${\pi }_a>0$, all real functions on $E$ belong to this finite-dimensional real inner-product space, denoted $L^2(E,\pi )$; equality in it is pointwise equality.

The associated jump-chain construction waits an exponential time of rate $q_a$ in state $a$ and then selects state $b\mathrm{\ne }a$ with probability $w_{ab}\mathrm{/}{q}_a$, using fresh independent random choices at successive jumps. Here an exponential time of rate $q>0$ has survival probability $\mathrm{Pr}(\tau >t)=e^{-qt}$ for $t\ge 0$. The Ionescu–Tulcea construction Ionescu-Tulcea construction of a Markov chain supplies the countable random sequence under the stated Choice assumption. The theorem on this page proves nonexplosion, identifies the transition semigroup, and establishes the positive reversible spectral gap. This is a finite mathematical rate model, not a derivation of rates from Hamiltonian mechanics.

Assume the Axiom of Choice The Axiom of Choice, the filtered Brownian probability-space convention of Ornstein–Uhlenbeck integral equation, and the adopted Newtonian spacetime and frame assumptions of Inertial frames, stationary observers and the adopted Galilean transformations between frames. Fix one inertial frame $\mathrm{\Gamma }=(o,u,(e_1,e_2,e_3))$ of that postulate: it is a fixed global adapted affine chart, with $o\in A$, $u\in V$, $\tau (u)=1$, and a fixed positively oriented $g$-orthonormal basis $(e_1,e_2,e_3)$ of the spatial displacement space $K=\ker \tau$. Its chart map assigns each event $p\in A$ the unique coordinates $(t_{\mathrm{\Gamma }}(p),{\mathbf{x}}_{\mathrm{\Gamma }}(p))$ satisfying $p=o+t_{\mathrm{\Gamma }}(p)u+{\sum }_{i=1}^3{x}_{\mathrm{\Gamma },i}(p)e_i$. Let $X=(X_t{)}_{t\ge 0}$ be the scalar position-coordinate process in this chart, measured in metres, whose value is the signed first spatial coordinate along the fixed axis $e_1$. The model tracks only $X$; it does not specify a full particle worldline or assert a law for either of the other spatial coordinates. Let $X_0$ be an ${\mathcal{F}}_0$-measurable real-valued random variable. On a measurable event ${\mathrm{\Omega }}_{\ast }\in \mathcal{F}$ with $P({\mathrm{\Omega }}_{\ast })=1$, assume $t\mapsto X_t(\omega )$ is continuously differentiable and define the velocity coordinate $V_t(\omega ):=d{X}_t(\omega )\mathrm{/}dt$ there, measured in metres per second. Thus $X_t=X_0+{\int }_0^t{V}_{s}ds$ on ${\mathrm{\Omega }}_{\ast }$; values on the null complement do not affect the stochastic conclusions.

Take a primitive particle mass $m>0$ in kilograms, a primitive constant bath temperature $T_b>0$ in kelvins, and the primitive SI Boltzmann constant $k_B>0$ in joules per kelvin. The positive damping rate $\gamma$ is a primitive model parameter in seconds${}^{-1}$; the positive noise amplitude $\sigma$ is derived as the square root of $2\gamma k_B{T}_b\mathrm{/}m$ and has units metres seconds${}^{-3\mathrm{/}2}$. Let $B_t$ be standard scalar Brownian motion with $B_0=0$, variance $t$ in seconds, and future increments independent of the present filtration. Prepare an ${\mathcal{F}}_0$-measurable $V_0\sim N(0,k_B{T}_b\mathrm{/}m)$ independently of future Brownian increments. The force $F:[0,\mathrm{\infty })\to \mathbb{R}$ is a prescribed deterministic locally continuous input in newtons. Adopt

$V_t=V_0-\gamma {\int }_0^t{V}_{s}ds+\frac{1}{m}{\int }_0^{t}F(s)ds+\sigma B_t,{\sigma }^2=\frac{2\gamma k_B{T}_b}{m}\mathrm{.}$

The displayed integral equation is the definition of the stochastic differential notation $d{V}_t=-\gamma V_{t}dt+F(t)dt\mathrm{/}m+\sigma d{B}_t$. Brownian increments have units seconds${}^{1\mathrm{/}2}$, so every term in the equation for $V_t$ has units metres seconds${}^{-1}$. When $F=0$, the preparation is the stationary OU Gaussian law because its variance is ${\sigma }^2\mathrm{/}(2\gamma )=k_B{T}_b\mathrm{/}m$; the cited mathematical supplier establishes the specified OU model properties. The thermal noise normalization, Gaussian preparation, constant damping, and deterministic force coupling are adopted assumptions. No measurement, material calibration, nonlinear response, or microscopic derivation of this bath is asserted.

Use the finite state set, energies $e_a$, rates $w_{ab}$, Gibbs weights ${\pi }_a$, and row generator $Q$ defined in Finite reversible stochastic rate model. Assign the state energies the physical meaning of coarse-state energies in joules. Use the primitive SI Boltzmann constant $k_B$ in joules per kelvin from Adopted inertial velocity Langevin bath, and set $\beta =(k_B{T}_b{)}^{-1}$ in joules${}^{-1}$ for the positive bath-temperature parameter $T_b$ in kelvins. Adopt the baseline transition rates $w_{ab}$ in seconds${}^{-1}$ and the initial law $p_0=\pi$ for the equilibrium-response branch. These choices define a finite phenomenological thermal rate model; they do not say that a real material has these rates.

Fix a finite time horizon $t_f>0$. For a fixed state observable $B:E\to \mathbb{R}$ with units denoted $[B]$, a dimensionless control amplitude $ϵ$, and a continuous field $h:[0,t_f]\to \mathbb{R}$ with units joules per $[B]$, adopt the controlled rates

$w_{ab}^{ϵ}(t)=w_{ab}\exp \text{\hspace{0.17em}⁣}(\frac{\beta ϵh(t)}{2}(B_b-B_a)),a\mathrm{\ne }b\mathrm{.}$

Their units remain seconds${}^{-1}$ because the exponent is dimensionless. This is one explicit symmetric allocation of the field-dependent rate factor. For constant $h$ and each positive-rate edge ($w_{ab}>0$, equivalently $w_{ba}>0$), the rate ratio obeys

$\frac{w_{ab}^{ϵ}}{w_{ba}^{ϵ}}=\exp \text{\hspace{0.17em}⁣}(-\beta [(e_b-ϵh{B}_b)-(e_a-ϵh{B}_a)]),$

so the Gibbs weights of the shifted energies $e_a-ϵh{B}_a$ satisfy detailed balance by direct multiplication on every positive-rate edge. For an absent edge, both controlled rates remain zero, so the detailed-balance product identity holds there as well; no rate ratio is defined there. A rate ratio alone leaves state-pair prefactors undetermined; the symmetric splitting above is an additional modeling choice and is the specific perturbation analyzed in the consuming theorem. No experiment or empirical rate estimate is asserted.

Assume the Axiom of Choice The Axiom of Choice and let $E,\pi ,w,Q$ be the finite model data of Finite reversible stochastic rate model, with $\mathrm{\mid }E\mathrm{\mid }=r\ge 2$. For every initial probability row vector $p_0$, there exists a time-homogeneous continuous-time Markov chain $X_t$ on $E$, $t\ge 0$, with no finite accumulation of jumps, transition matrix $P_t=e^{tQ}$, and law $p_t=p_0{P}_t$. The detailed-balance law $\pi$ is stationary. On the real inner-product space $L^2(E,\pi )$, $Q$ is self-adjoint and has an orthonormal eigenbasis ${\varphi }_0=1,{\varphi }_1,\dots ,{\varphi }_{r-1}$ with eigenvalues $0,-{\lambda }_1,\dots ,-{\lambda }_{r-1}$, where every ${\lambda }_j>0$. Thus the spectral gap ${\lambda }_{\ast }:={\min }_j{\lambda }_j$ is positive and $P_t$ converges exponentially to projection onto constants.

Here a finite-state continuous-time Markov chain means a process with piecewise-constant paths whose conditional transition matrix over an interval of length $t$ is $P_t$, independent of the starting time. For initial law $\pi$, the chain has a two-sided stationary extension $X_t$, $t\in \mathbb{R}$, whose finite-dimensional laws at $t_1<\cdots <t_n$ are ${\pi }_{a_1}{\prod }_{i=1}^{n-1}{P}_{t_{i+1}-t_i}(a_i,a_{i+1})$. Its sample paths are locally piecewise constant with finitely many jumps on every bounded time interval.

Assume the Axiom of Choice The Axiom of Choice and the Brownian probability-space hypotheses in Ornstein–Uhlenbeck integral equation. Let $\gamma ,m,\mathrm{\Theta },k_B>0$, $\sigma >0$, and ${\sigma }^2=2\gamma k_B\mathrm{\Theta }\mathrm{/}m$. Here these are real mathematical parameters; a separate physics postulate assigns their physical units and interpretation. Let $V_0:\mathrm{\Omega }\to \mathbb{R}$ be ${\mathcal{F}}_0$-measurable, satisfy $V_0\sim N(0,k_B\mathrm{\Theta }\mathrm{/}m)$ and be independent of future Brownian increments. For any locally continuous deterministic input $u:[0,\mathrm{\infty })\to \mathbb{R}$, and every finite $t_f>0$, the equation restricted to $[0,t_f]$ $V_t=V_0-\gamma {\int }_0^t{V}_{s}ds+\frac{1}{m}{\int }_0^{t}u(s)ds+\sigma B_t,0\le t\le t_f,$ has a unique continuous solution; these unique finite-horizon solutions agree on overlapping intervals and define one solution for all $t\ge 0$. Its mean is $\mathbb{E}{V}_t=\frac{1}{m}{\int }_0^t{e}^{-\gamma (t-s)}u(s)ds\mathrm{.}$ For $u=0$, $V$ is a stationary centered Gaussian process on $[0,\mathrm{\infty })$ with covariance $\mathbb{E}[V_s{V}_t]=v_{\mathrm{\Theta }}{e}^{-\gamma \mathrm{\mid }t-s\mathrm{\mid }},v_{\mathrm{\Theta }}:=k_B\mathrm{\Theta }\mathrm{/}m\mathrm{.}$ Define $Y_t:={\int }_0^t{V}_{s}ds$ and $D:=v_{\mathrm{\Theta }}\mathrm{/}\gamma$. Then $\mathrm{Var}(Y_t)=2v_{\mathrm{\Theta }}(\frac{t}{\gamma }-\frac{1-e^{-\gamma t}}{{\gamma }^2}),{\lim }_{t\to \mathrm{\infty }}\frac{\mathrm{Var}(Y_t)}{2t}=D\mathrm{.}$ For every deterministic input, the centered displacement $Y_t-\mathbb{E}{Y}_t$ is Gaussian with the variance above; consequently $(Y_t-\mathbb{E}{Y}_t)\mathrm{/}\sqrt{t}$ converges in distribution to $N(0,2D)$. For constant input $u(s)=u_0$, ${\lim }_{t\to \mathrm{\infty }}\mathbb{E}[V_t]\mathrm{/}{u}_0=\chi :=1\mathrm{/}(m\gamma )$ when $u_0\mathrm{\ne }0$, and $D=\chi k_B\mathrm{\Theta }$.

Assume the adopted velocity bath Adopted inertial velocity Langevin bath on $t\ge 0$ and the Axiom of Choice The Axiom of Choice. Let $X_t=X_0+{\int }_0^t{V}_{s}ds$ be the position coordinate in that model, and prepare $V_0$ in its specified stationary Gaussian law. For zero external force, define the stationary velocity covariance by $C_{VV}(\tau ):=\mathbb{E}[(V_s-\mathbb{E}{V}_s)(V_{s+\tau }-\mathbb{E}{V}_{s+\tau })],s\ge \max (0,-\tau )\mathrm{.}$ Stationarity makes it independent of $s$, and the model gives $C_{VV}(\tau )=\frac{k_B{T}_b}{m}{e}^{-\gamma \mathrm{\mid }\tau \mathrm{\mid }}\mathrm{.}$ For a deterministic force applied at time $s\le t$, the linear response kernel is $R_{VF}(t-s)=\frac{1}{m}{e}^{-\gamma (t-s)}=\frac{1}{k_B{T}_b}{C}_{VV}(t-s)\mathrm{.}$ Here $R_{VF}$ is defined by the exact mean-response formula $\mathbb{E}[V_t]={\int }_0^t{R}_{VF}(t-s)F(s)ds$ for deterministic forcing from the stated zero-mean stationary preparation; its units are $\mathrm{k}{\mathrm{g}}^{-1}$. The constant-force mobility below is a distinct steady-response coefficient. The diffusion coefficient defined by centered displacement variance satisfies the Green–Kubo identity $D:={\lim }_{t\to \mathrm{\infty }}\frac{\mathrm{Var}(X_t-X_0)}{2t}=\frac{k_B{T}_b}{m\gamma }\text{in }{\mathrm{m}}^2{\mathrm{s}}^{-1}\mathrm{.}$ More precisely, $D={\int }_0^{\mathrm{\infty }}{C}_{VV}(\tau )d\tau$. The constant-force mobility $\mu :={\lim }_{t\to \mathrm{\infty }}\frac{\mathbb{E}[V_t]}{F_0}=\frac{1}{m\gamma }\text{in }\mathrm{s}\mathrm{k}{\mathrm{g}}^{-1}$ for constant $F_0\mathrm{\ne }0$. Hence $D=\mu k_B{T}_b$. The centered displacement divided by $\sqrt{t}$ converges in distribution to $N(0,2D)$. These are conditional model deductions, not observed transport data or a universal law for arbitrary systems.

The selected mean-field definition Smooth periodic finite-N and scaled mean-field phase data fixes a smooth periodic pair force and $1\mathrm{/}N$ interaction scaling. The collision definition Cutoff elastic-reflection collision measure and gain fixes a homogeneous finite normalized angular reflection measure. The finite-rate definition Finite reversible stochastic rate model fixes a connected finite state graph and reversible rates. These formulations provide the precise regimes used by the results on this page.

No result here derives the finite-state rates from deterministic particle mechanics, proves a limit from particle dynamics to continuum fluid equations, or shows that long-time averages equal invariant-measure averages for a general Hamiltonian system. Such conclusions require their own stated microscopic model, limiting assumptions and mathematical proof. The Langevin model here is exactly Adopted inertial velocity Langevin bath: a scalar velocity process with positive constant mass, damping rate and bath temperature, additive Brownian forcing, the adopted thermal noise normalization, stationary Gaussian initial velocity, and prescribed deterministic continuous force, satisfying its displayed integral equation. The physical rate and Langevin models in this pair are adopted assumptions; their theorems are conditional predictions, and this pair contains no reported observations or empirical inference.

Assume the Axiom of Choice The Axiom of Choice, countable choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), and the rate model $E,\pi ,w,Q$ of Finite reversible stochastic rate model. Let $X_t$ be the stationary two-sided chain of Nonexplosive finite continuous-time chain and reversible spectral gap, with transition semigroup $P_t=e^{tQ}$. For any real observables $A,B:E\to \mathbb{R}$, write $A_c=A-\pi (A)$ and $B_c=B-\pi (B)$, and define $C_{BA}(t):=⟨B_c,P_t{A}_c{⟩}_{\pi },t\ge 0.$ Define ${\mathrm{Cov}}_{\pi }(A,B):=\pi (AB)-\pi (A)\pi (B)$, where $AB$ is pointwise multiplication. For any finite horizon $t_f>0$, let $h:[0,t_f]\to \mathbb{R}$ be continuous and let $ϵ$ be dimensionless. For $0\le t\le t_f$, prescribe rates $w_{ab}^{ϵ}(t)=w_{ab}\exp \text{\hspace{0.17em}⁣}(\frac{\beta ϵh(t)}{2}(B_b-B_a)),a\mathrm{\ne }b,$ with negative row sums on the diagonal, and let $p^{ϵ}(0)=\pi$ evolve by the resulting finite forward equation. For $0\le t\le t_f$, the derivative of the observable mean at $ϵ=0$ is ${\frac{\mathrm{\partial }}{\mathrm{\partial }ϵ}{\sum }_a{p}_a^{ϵ}(t)A_a\mid }_{ϵ=0}={\int }_0^{t}h(s)R_{AB}(t-s)ds,R_{AB}(u)=-\beta \frac{d}{du}{C}_{BA}(u)\mathrm{.}$ For a constant field $h$, define the normalized perturbed Gibbs vector by ${\pi }_a^{ϵ}:=\frac{{\pi }_a{e}^{\beta ϵh{B}_a}}{{\sum }_{c\in E}{\pi }_c{e}^{\beta ϵh{B}_c}}\mathrm{.}$ It is stationary and its static susceptibility is $\beta h{\mathrm{Cov}}_{\pi }(A,B)$.

For any bounded vector observable $G:E\to {\mathbb{R}}^d$, set $G_c=G-\pi (G)$ and $C_{ij}(s):=⟨(G_c{)}_i,P_s(G_c{)}_j{⟩}_{\pi }\mathrm{.}$ These correlations are absolutely integrable. The Green–Kubo covariance limit is ${\lim }_{T\to \mathrm{\infty }}\frac{{\mathrm{Cov}}_{\pi }({\int }_0^{T}G(X_s)ds)}{2T}=D,D_{ij}={\int }_0^{\mathrm{\infty }}{C}_{ij}(s)ds=⟨(G_c{)}_i,(-Q{)}^{-1}(G_c{)}_j{⟩}_{\pi },$ where $(-Q{)}^{-1}$ is the inverse on the mean-zero subspace and $D$ is positive semidefinite.