---
title: Harmonic Process in Nonequilibrium Systems
url: https://www.emergentmind.com/topics/harmonic-process
type: topic
---

# Harmonic Process in Nonequilibrium Systems

The **harmonic process** is an exactly solvable boundary-driven continuous-time Markov model on a one-dimensional chain with unbounded site occupancy, developed in nonequilibrium statistical mechanics and integrable probability as a tractable non-equilibrium steady-state (NESS) system [2507.20436]. In its algebraic formulation, the dynamics are written in “Hamiltonian” form and depend on a spin label \(2s\in\mathbb N\); in the boundary-driven setting, the stationary law admits several exact representations, including a closed gamma-product formula, a nested-integral or mixed-measure form, a matrix-product ansatz, and a mixture of product geometric measures [2507.20436], [2604.17469]. These representations support a detailed theory of local equilibrium, laws of large numbers, central limit theorems, and large deviations for local observables in a non-reversible interacting system [2604.17469].

## 1. Definition of the model and its dynamics

In the formulation emphasized in the steady-state analysis, the state space consists of configurations
\[
|m_1,\dots,m_N\rangle,\qquad m_i\in\mathbb N,
\]
on a chain of length \(N\), with unbounded occupancy at each site. The process is a continuous-time Markov chain whose generator is written as
\[
H=\mathcal B_L+\sum_{i=1}^{N-1}\mathcal H_{i,i+1}+\mathcal B_R,
\]
acting on
\[
|P\rangle=\sum_{\vec m}P(\vec m)|m_1,\dots,m_N\rangle,
\]
and the steady state satisfies \(H|\mu\rangle=0\) [2507.20436].

The bulk interaction between neighboring sites is governed by the harmonic weights
\[
h_s(m)=\sum_{k=1}^m \frac{1}{2s+k-1},
\]
and the jump kernel
\[
\phi_s(m,k)=\frac1k\cdot \frac{\Gamma(m+1)\Gamma(m-k+2s)}{\Gamma(m-k+1)\Gamma(m+2s)}.
\]
The local bulk operator acts by redistributing \(k\) particles from one site to its neighbor, with rates determined by \(\phi_s\). Boundary reservoirs inject according to parameters \(\beta_L,\beta_R\in(0,1)\) and extract with matching harmonic rates. The parameter \(2s\in\mathbb N\) controls both the bulk-jump spectrum and the harmonic weights [2507.20436].

A boundary-driven formulation used for ergodic analysis is written on
\[
\Omega_N=\mathbb N_0^{\{1,\dots,N\}},
\]
with configuration \(\eta=(\eta_1,\dots,\eta_N)\). There the bulk dynamics replace a pair \((\eta_i,\eta_{i+1})\) by a uniformly chosen split of the conserved total \(k=\eta_i+\eta_{i+1}\), while the left and right reservoirs replace the boundary occupation by geometric laws \(\nu_{\theta_L}\) and \(\nu_{\theta_R}\), respectively. The bulk conserves total mass, whereas the reservoirs exchange mass with the environment and create a genuine NESS when \(\theta_L\neq \theta_R\) [2604.17469].

## 2. Exact stationary measure

The stationary state is explicitly computable. In the quantum-inverse-scattering treatment, the unique normalized zero-mode can be represented as two nonlocal rotations acting on a telescopic product vector, with reservoir variables
\[
\rho_{L,R}=\frac{\beta_{L,R}}{1-\beta_{L,R}}.
\]
The resulting stationary weight can be written in fully explicit form as
\[
\pi(\vec m)\equiv \mu(\vec m)
= Z_N^{-1}\,(\rho_L-\rho_R)^{|\vec m|}
\prod_{i=1}^N \kappa(m_i)\,
\frac{\Gamma\!\left(2s(N+1-i)+\sum_{k=i}^N m_k\right)}
{\Gamma\!\left(2s(N-i+2)+\sum_{k=i}^N m_k\right)},
\]
where
\[
|\vec m|=\sum_{i=1}^N m_i,\qquad
\kappa(m)=\frac{\Gamma(2s+m)}{\Gamma(2s)\Gamma(m+1)},
\]
and
\[
Z_N=(\rho_L-\rho_R)^{-2s(N+1)+1}\cdot \frac{\Gamma(2s(N+1))}{\Gamma(2s)}.
\]
This is the closed-form formula derived by Frassek et al. [2107.01720].

For the boundary-driven harmonic model treated probabilistically, the NESS can also be expressed as a mixture of product geometric laws. Writing
\[
\nu_\theta(n)=(1+\theta)^{-1}\Bigl(\frac{\theta}{1+\theta}\Bigr)^n,\qquad n=0,1,2,\dots,
\]
with mean \(E_{\nu_\theta}[n]=\theta\), one has
\[
\mu_N(d\eta)=\int_{[0,\infty)^N}\bigotimes_{i=1}^N \nu_{\theta_i}(d\eta_i)\;
\Lambda_N(d\theta_1,\dots,d\theta_N).
\]
In the harmonic NESS, the random vector \((\Theta_{1,N},\dots,\Theta_{N,N})\) is distributed as the order statistics of \(N\) independent Uniform\([\theta_L,\theta_R]\) variables, equivalently
\[
\Theta_{i,N}=\theta_L+(\theta_R-\theta_L)\,U_{i:N},
\]
where \(U_{1:N}\le \cdots \le U_{N:N}\) are the usual order statistics. Conditional on \(\Theta\), the sites are independent with laws \(\nu_{\Theta_{i,N}}\) [2604.17469].

## 3. Three exact steady-state representations

A central structural result is that the harmonic-process NESS admits three equivalent exact representations: a closed gamma-product formula, a nested-integral representation, and a matrix-product ansatz [2507.20436], [2307.02793], [2307.14975].

Carinci–Giardinà et al. rewrote the stationary weight as an \(N\)-fold iterated integral over auxiliary variables \(\theta_i\in[\rho_L,\rho_R]\):
\[
\pi(\vec m)
= \frac{1}{(\rho_R-\rho_L)^{2s(N+1)-1}\,[\Gamma(2s(N+1))/\Gamma(2s)^{N+1}]}
\]
\[
\times \int_{\rho_L}^{\rho_R} d\theta_1
\int_{\theta_1}^{\rho_R} d\theta_2 \cdots
\int_{\theta_{N-1}}^{\rho_R} d\theta_N
\prod_{i=1}^{N+1}(\theta_i-\theta_{i-1})^{2s-1}
\prod_{i=1}^N
\kappa(m_i)\Bigl[\frac{\theta_i}{1+\theta_i}\Bigr]^{m_i}(1+\theta_i)^{-2s},
\]
with \(\theta_0:=\rho_L\) and \(\theta_{N+1}:=\rho_R\). This representation gives the stationary state a mixed-measure interpretation in terms of auxiliary reservoir variables [2307.02793], [2307.14975].

The new contribution of the matrix-product treatment is an explicit oscillator realization. One seeks operators \(X(m)\) and boundary vectors \(\langle V|\), \(|W\rangle\) such that
\[
\pi(m_1,\dots,m_N)=Z_N^{-1}\,\langle V|X(m_1)\cdots X(m_N)|W\rangle.
\]
Using oscillators \(a,a^\dagger\) with \([a,a^\dagger]=1\) and Fock basis \(|k\rangle\), an explicit choice is
\[
X(m)_{a,b}
=\langle a|X(m)|b\rangle
=\kappa(m)\,\delta_{a,\,2s+m+b}\,
\frac{\Gamma(m+b+1)}{\Gamma(2s+m+b+1)},
\]
with boundary vectors
\[
\langle V|a\rangle=(\rho_L-\rho_R)^a,\qquad
\langle a|W\rangle=\delta_{a,\,2s-1}.
\]
For this choice, the matrix product reproduces the closed gamma-product weight exactly [2507.20436].

The relative roles of the three representations are distinct.

| Representation | Core form | Typical use |
|---|---|---|
| Closed form | Telescopic gamma-product | Asymptotic analysis, large-charge limits |
| Nested integrals | \(N\)-fold mixed-measure integral | Functional identities, correlation integral formulas |
| Matrix-product ansatz | Infinite-dimensional oscillator MPA | Correlation functions, algebraic steady-state computations |

The closed form is the most explicit. The nested-integral form is flexible for functional relations in the reservoir parameters. The MPA is the most algebraic, naturally connected with the integrability framework and direct expectation-value computations [2507.20436].

## 4. Transformations between representations

The equivalence of the three steady-state formulas is itself an exact structural result. From the closed gamma-product formula, one obtains the nested-integral representation by repeatedly applying the Beta-function identity
\[
\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}
=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}\,dt.
\]
Each ratio of gamma functions is converted into a one-fold integral; after \(N\) iterations, the telescopic product becomes an \(N\)-fold nested integral [2507.20436].

The passage from nested integrals to the MPA is mediated by an integral operator
\[
(A_m f)(t)=\frac{\kappa(m)}{\Gamma(2s)}\int_0^t f(r)\,r^m\,(t-r)^{2s-1}\,dr.
\]
Interpreting \(A_m\) as an infinite matrix yields entries coinciding with \(X(m)_{a,b}\), and the full nested integral becomes
\[
\langle V|A_{m_1}A_{m_2}\cdots A_{m_N}|W\rangle.
\]
Conversely, the MPA recovers the closed form either by diagonalizing the oscillators or by computing \(\langle V|X(m_1)\cdots X(m_N)|W\rangle\) through successive action on the Fock basis, which reproduces the gamma-product expression [2507.20436].

At the algebraic level, stationarity is encoded by a bulk relation and matching boundary relations involving \(X(m)\) and an auxiliary family \(\bar X(m)\). In the summary form,
\[
X(m)\bar X(m')-\bar X(m)X(m')
= (\mathcal H\bullet \text{ on }X\otimes X),
\]
together with left and right boundary identities for \(\mathcal B_L\) and \(\mathcal B_R\). These relations guarantee \(H|\mu\rangle=0\) and place the harmonic process within the same matrix-product and Zamolodchikov-algebra paradigm that appears in other integrable boundary-driven systems [2507.20436].

## 5. Local equilibrium, fluctuations, and large deviations

Because the stationary measure is an explicit mixture of product states, the harmonic process admits a detailed ergodic theory. The local-equilibrium profile is linear:
\[
\rho(x)=\theta_L+x(\theta_R-\theta_L),\qquad x\in[0,1].
\]
For every fixed multi-index \(p=(p_1,\dots,p_k)\in\mathbb N^k\) and \(x\in(0,1)\),
\[
E_{\mu_N}\!\left[\eta_{\lfloor xN\rfloor+1}^{p_1}\cdots \eta_{\lfloor xN\rfloor+k}^{p_k}\right]
\to
\prod_{j=1}^k E_{\nu_{\rho(x)}}[n^{p_j}]
\]
almost surely. A quantitative concentration bound for the order-statistic parameters is
\[
P\!\left(\max_{1\le i\le N} |\Theta_{i,N}-E\Theta_{i,N}|\ge \varepsilon\right)\le \frac{1}{N\varepsilon^2}.
\]
For self-duality polynomials \(D_N\), the deviation from local equilibrium is controlled at order \(O(1/N)\) uniformly in \(x\) [2604.17469].

For a local observable \(g\) of range \(k\) and a test function \(\phi\in C[0,1]\), the empirical field
\[
X_N(g;\phi)=\frac1N\sum_{i=0}^{N-k}\tau_i g(\eta)\,\phi\!\left(\frac{i}{N+1}\right)
\]
satisfies a strong law of large numbers:
\[
X_N(g;\phi)\to \int_0^1 h(\rho(x))\,\phi(x)\,dx,
\qquad
h(\rho)=E_{\bigotimes_{i=1}^k \nu_\rho}[g(\eta_1,\dots,\eta_k)].
\]
The proof decomposes \(\tau_i g\) into its conditional expectation given \(\Theta\) and a centered remainder; the first part is handled as an \(L\)-statistic in order statistics, the second as a triangular array with uniform moment bounds [2604.17469].

The centered fluctuation field
\[
Y_N(g;\phi)=\frac1{\sqrt N}\sum_{i=0}^{N-k}
\bigl[\tau_i g(\eta)-E_{\mu_N}[\tau_i g(\eta)]\bigr]
\phi\!\left(\frac{i}{N+1}\right)
\]
converges in law to a Gaussian with variance
\[
\sigma_T^2+\sigma_E^2.
\]
Here
\[
\sigma_E^2=\int_0^1 V(\rho(x))\,\phi(x)^2\,dx
\]
is the equilibrium white-noise contribution, while
\[
\sigma_T^2=(\theta_R-\theta_L)^2
\int_0^1\!\!\int_0^1
[s\wedge t-st]\,
\phi(s)\phi(t)\,
h'(\rho(s))h'(\rho(t))\,ds\,dt
\]
comes from fluctuations of the random environment \(\Theta_{i,N}\), which converge to a Brownian bridge. For the density field \(g(\eta)=\eta_1\), the limit splits into a Brownian-bridge term plus centered white noise [2604.17469].

Large deviations are also explicit. For
\[
\Xi(N,g)=\frac1N\sum_{i=0}^{N-k}\tau_i g(\eta)\,\delta_{i/(N+1)},
\]
one obtains an LDP at speed \(N\). The random parameter profile \(\Theta_N(\cdot)\) has rate function
\[
J(\theta(\cdot))
=
\int_0^1
\left[-\log\!\left(\frac{\theta'(t)}{\theta_R-\theta_L}\right)\right]dt
\]
when \(\theta(\cdot)\) is absolutely continuous, strictly increasing, and satisfies \(\theta(0)=\theta_L\), \(\theta(1)=\theta_R\), and \(J=\infty\) otherwise. The final rate function for the mixed measure is the variational formula
\[
\mathcal I(\mu)=
\inf_{\theta(\cdot)}
\left\{
\int_0^1 I(\theta(x),\mu(x);g)\,dx
+
J(\theta(\cdot))
\right\}.
\]
This yields LLN, CLT, and LDP within a single exact framework [2604.17469].

## 6. Analytical significance and methodological role

The harmonic process occupies a distinctive position because its NESS is simultaneously accessible by integrable, probabilistic, and operator-algebraic methods. On the integrability side, the closed-form stationary state was obtained by the quantum-inverse-scattering method, the nested-integral representation exposes a mixed-measure structure, and the matrix-product ansatz furnishes a new oscillator-based auxiliary-space realization that had not previously been available for the model [2107.01720], [2307.02793], [2307.14975], [2507.20436].

On the probabilistic side, the mixture-of-products structure reduces fluctuation questions to two components: the product-measure part and the random profile formed by order statistics. The proofs then combine empirical-process theory for order statistics, strong laws and central limit theorems for triangular arrays, a positive maximum-principle perspective, self-duality, and Varadhan’s lemma [2604.17469]. In this sense, the harmonic process provides a complete microscopic derivation of macroscopic fluctuation results for a non-reversible, boundary-driven interacting system [2604.17469].

The three exact stationary representations are complementary rather than redundant. The closed form is particularly suited to asymptotic analysis as \(N\to\infty\) or in large-charge regimes. The nested-integral form is adapted to functional identities and correlation-function integral formulas. The matrix-product ansatz is adapted to algebraic computations of observables, including 1- and 2-point functions, without combinatorial explosion, even though its auxiliary space is infinite-dimensional and realized on oscillator Fock space [2507.20436].

Taken together, these results define the harmonic process as a rare exactly solvable nonequilibrium lattice model for which stationary structure, algebraic representation, and asymptotic fluctuation theory can all be developed explicitly and in mutually compatible forms.

Source: https://www.emergentmind.com/topics/harmonic-process