---
title: Time-Dependent Glauber Rates
url: https://www.emergentmind.com/topics/time-dependent-glauber-rates
type: topic
---

# Time-Dependent Glauber Rates

Time-dependent Glauber rates define the stochastic rules governing the evolution of spin systems, generalizing the classic Glauber dynamics to contexts where rates either depend on local time-dependent fields, spatial bias, or allow only approximate compliance with statistical physical constraints such as detailed balance. These rates play a central role in nonequilibrium statistical mechanics, enabling the study of driven Ising chains, single-electron quantum states, and the derivation of linear dynamical field theories. Their significance lies in their ability to interpolate between equilibrium relaxation governed by detailed balance and far-from-equilibrium scenarios controlled by global balance, providing a direct link between microscopic dynamics and macroscopic, often exactly solvable, phenomenology.

## 1. Definition and Functional Forms

Time-dependent Glauber rates originate from the extension of Markovian spin-flip dynamics, where the probability per unit time $w_n(\sigma,t)$ for a spin $\sigma_n = \pm 1$ to flip depends not only on its local neighborhood but also on explicit time-dependent parameters or external fields.

**Generalized rate (Ising chain, one-parameter family):**
$$
w_n(\sigma) = \frac{1}{2} \alpha [1 - \gamma\,\sigma_n (p\,\sigma_{n-1} + (1-p)\,\sigma_{n+1})]
$$
with $\alpha$ as time-scale, $\gamma = \tanh(2J/T)$ (enforcing consistency with the Ising coupling $J$ and temperature $T$), and $p\in [0,1]$ tuning spatial bias; $V = 2p-1$ provides a drift parameterization [1102.0141].

**With external time-dependent field $h_n(t)$:**
$$
w_n(\sigma, t) = \frac{1}{2} \alpha [1 - \gamma\,\sigma_n (p\,\sigma_{n-1} + (1-p)\,\sigma_{n+1})][1 - \kappa_n(t)\sigma_n]
$$
where $\kappa_n(t) = \tanh[h_n(t)/T]$. Here, explicit time-dependence arises solely from $\kappa_n(t)$, not from the bias itself [1102.0141].

**Optimal linearization (arbitrary graph):**
$$
w_j(\sigma_j) = \frac{\alpha}{2}\Bigl[1 + \sum_{k=1}^z \gamma_k\,\sigma_j \sigma_k\Bigr]
$$
with coefficients $\gamma_k$ determined via Moore–Penrose regression to minimize detailed-balance violation [1401.5412]. In isotropic cases, $\gamma_k \equiv \gamma$ admits a closed form as a function of temperature ($\beta$), coupling ($J$), and coordination ($z$).

## 2. Conditions Imposed by Global and Detailed Balance

When the dynamics is symmetric ($p=1/2$), Glauber’s original rate satisfies detailed balance with respect to the Ising Gibbs measure:
$$
w_n(\sigma) = \frac{1}{2}\alpha[1 - \frac{\gamma}{2} \sigma_n (\sigma_{n-1} + \sigma_{n+1})]
$$
This ensures reversibility on each bond [1102.0141].

For generic $p\neq 1/2$ (“directed” or “asymmetric” cases), detailed balance fails. Global balance is imposed instead. The master equation, summed over all possible flips and weighted by the Ising-Gibbs stationary measure, requires:
$$
\sum_n \{ w_n(\sigma) - w_n(\sigma^n) (P_{st}(\sigma^n)/P_{st}(\sigma)) \} = 0,
$$
where $\sigma^n$ denotes the configuration with spin $n$ flipped, and $P_{st}(\sigma) \propto \exp[-\beta J \sum_n \sigma_n \sigma_{n+1}]$ [1102.0141]. Detailed algebra constrains $\gamma = \tanh 2K$ (with $K=J/T$) and yields the unique globally balanced rate for fully directed cases ($p=0$ or $1$):
$$
w_n(\sigma) = \frac{1}{2}\alpha[1 - \gamma \sigma_n \sigma_{n\pm1}]
$$
Up to $\alpha$, this is uniquely set by global balance alone [1102.0141].

## 3. Manifestation of Time Dependence

Time dependence of Glauber rates emerges primarily through external, space- and time-dependent magnetic fields. In zero applied field, transition probabilities remain purely configuration-dependent (no explicit $t$). Under $h_n(t)$, rates gain explicit $t$-dependence via $\kappa_n(t)$.

In kinetic Ising models, this means:
- **Without field:** Coarsening (aging) or stationary regime rates depend only on $\sigma$.
- **With $h_n(t)$:** $w_n(\sigma, t)$ encodes explicit temporal variation, relevant for linear response and nonequilibrium driving [1102.0141, 1401.5412].

For the linearized (optimal) forms relevant for analytic solution, the field-dependence causes the relaxation rates and susceptibilities to become explicit functions of $H(t)$, with quadratic corrections in weak fields [1401.5412].

## 4. Symmetric, Asymmetric, and Continuum Limits

Time-dependent Glauber rates bridge between symmetric, detailed-balance-preserving, and asymmetric, globally-balanced forms:

| Case                | Explicit Rate Formula                                             | Balance Condition     |
|---------------------|------------------------------------------------------------------|-----------------------|
| $p=1/2$ (symmetric) | $\frac{1}{2}\alpha [1 - \frac{\gamma}{2}\sigma_n(\sigma_{n-1}+\sigma_{n+1})]$ | Detailed balance      |
| $p=0$ or $1$ (directed) | $\frac{1}{2}\alpha [1 - \gamma\sigma_n\sigma_{n\pm1}]$                  | Global balance only   |
| $0<p<1$ (intermediate) | Interpolates via $p$ in the general rate                                 | Interpolates between  |

The continuous interpolation affects key dynamical observables:
- For $|V| < V_c = \sqrt{1-\gamma^2}$, correlation and response decay rates remain identical; the limit fluctuation-dissipation ratio ($X_\infty$) decreases from $1$ (equilibrium) to $0$ at the critical bias [1102.0141].
- Beyond $V_c$, the response decays faster, and $X_\infty = 0$.

In the continuum (linearized) limit, time-dependent Glauber rates give rise to the time-dependent Ginzburg–Landau (TDGL) equation:
$$
\frac{\partial m(\mathbf{r},t)}{\partial t} = \frac{1}{2}D\nabla^2 m(\mathbf{r},t) - \frac{1}{2}\kappa m(\mathbf{r},t)
$$
with $D=2\alpha |\gamma|$ and $\kappa=2\alpha(1-2d|\gamma|)$, directly connecting microscopic rates with macroscopic relaxation and pattern formation [1401.5412].

## 5. Observable Consequences: Magnetization, Correlation, Response, and Fluctuation-Dissipation

**Magnetization:** The evolution equation in the directed Ising chain takes the form:
$$
\frac{dM_n}{dt} = -M_n + \gamma[p M_{n-1} + (1-p) M_{n+1}]
$$
with solutions obtainable via Fourier-Laplace analysis. The bias parameter $V$ modifies propagation, introducing a drift term [1102.0141].

**Equal-time correlations:** The equation
$$
\frac{\partial C_n}{\partial t} = -2C_n + \gamma(C_{n-1} + C_{n+1})
$$
is independent of $p$ and $V$; thus, the domain growth law and scaling at $T=0$ coincide with the symmetric case. This suggests bias affects dynamic but not static spatial structures [1102.0141].

**Two-time correlations and response:** Asymmetric rates generate nontrivial temporal correlations with decay characteristics crossing over at the critical bias $V_c$. The linear response function satisfies a generalized fluctuation-dissipation relation:
$$
R_n(s,t) = \frac{1}{2}[\partial_s C_n(s,t) - \partial_t C_n(s,t)] - \frac{1}{2}\gamma V[C_{n+1}(s,t) - C_{n-1}(s,t)]
$$
Reducing to the equilibrium FDT at $V=0$, this highlights the central dynamical alteration induced by directedness [1102.0141].

**Relaxation and fluctuation-dissipation theorem (LGM):** In the linear Glauber model, the exact relaxation time in uniform field $H$ is:
$$
\tau_H^{-1} = \alpha[1 + z\gamma(T,H)],\quad \tau_H^{-1} = \tau^{-1} - \mathrm{sgn}(J)\alpha\beta^2 F H^2 + O(H^4)
$$
showing quadratic $H$-dependence [1401.5412]. The fluctuation-dissipation relation is preserved despite approximate balance conditions:
$$
\int_{-\infty}^{\infty} \langle \Sigma(0)\Sigma(t) \rangle e^{i\omega t}dt = \frac{2k_BT}{\omega}\mathrm{Im}\,\chi(\omega)
$$

## 6. Quantum and Single-Particle Generalizations

Time-dependent Glauber rates also underpin the theory of single-electron quantum sources, where the relevant quantities are the first-order and second-order (Glauber) correlation functions:
$$
G^{(1)}(t_1, t_2) = \langle \psi^\dagger(t_2)\psi(t_1)\rangle
$$
For periodically driven sources, this is formulated using Floquet theory. In the adiabatic case, $G^{(1)}$ is a symmetric Lorentzian in time, with lifetime $T_1$ and coherence time $T_2 = 2T_1$—the Fourier-transform-limited regime. In non-adiabatic pumping, $G^{(1)}$ becomes asymmetric and decays exponentially. The emission probability rate $G^{(1)}(t,t)$ connects directly to experimentally measurable current pulses [1212.0088].

A pure on-demand single-electron state ensures $G^{(2)}(t_1,t_2)$ vanishes, realizing full anti-bunching and substantiating the quantumness of the source [1212.0088].

## 7. Implications and Applications

Time-dependent Glauber rates establish a rigorous framework for analyzing nonequilibrium processes in spin systems, stochastic processes on networks, and quantum sources. Their flexibility enables the study of non-reversible dynamics (directed Ising chains [1102.0141]), analytic derivation of mean-field and continuum kinetic equations (linear and optimal LGM [1401.5412]), and fully quantum-coherent time-dependent transport (Floquet-driven sources [1212.0088]).

A notable implication is that nontrivial time dependence (in the absence of external field) arises not from spatial bias, but from the explicit introduction of a time-varying field or drive. This distinction clarifies the terminology: “time-dependent Glauber rates” conventionally refers to rates gaining $t$-dependence through an external field, not merely through asymmetric neighbor coupling [1102.0141].

Source: https://www.emergentmind.com/topics/time-dependent-glauber-rates