---
title: Log-Size Markov Jump Generator
url: https://www.emergentmind.com/topics/markov-jump-generator-in-log-size
type: topic
---

# Log-Size Markov Jump Generator

A Markov jump generator in log-size is an exact, stochastic operator governing the evolution of particle size distributions under pure-breakage processes, formulated after a change of variables to the logarithm of particle size. Derived from the deterministic population balance equations (PBEs), this generator acts on the mass-weighted, tagged-mass distribution over log-sizes and encapsulates both the breakage rate and the law for jumps in log-size, both set by the underlying fragmentation kernel. The framework is generically non-self-adjoint, admits a Lindblad embedding, and is extensible to non-Hermitian quantum field theory (NHQFT) via second quantization, providing access to fluctuations, size-size correlations, non-equilibrium universality, and explicit solutions in special cases such as the Airy quadratic sector [2601.06635].

## 1. Population Balance Equations and Kernel Inputs

The starting point is the deterministic PBE for particle number density $f(x, t)$, $x>0$, which under pure-breakage reads:
\[
\partial_t f(x,t)
= - S(x) f(x, t)
+ \int_x^\infty S(y) b(x, y) f(y, t) \,dy
\]
Here, the breakage (selection) rate $S(x)\geq 0$ and the daughter-number kernel $b(x, y)\geq 0$ encode, respectively, the intensity of fragmentation for particles of size $x$ and the expected number density of fragments of size $x$ produced from a parent of size $y$. Mass conservation is enforced by
\[
\int_0^y x\, b(x, y) \,dx = y
\]
implying that total mass $M = \int_0^\infty x f(x, t)\,dx$ remains invariant. Homogeneous kernels adopt the form
\[
S(x) = k x^\alpha, \qquad b(x, y) = \frac{1}{y} B\left(\frac{x}{y}\right),\,\,\, 0 < x < y
\]
with $k>0$, $\alpha \in \mathbb{R}$, and $B(z)$ a normalized dimensionless daughter density on $z\in(0, 1)$ satisfying
\[
\int_0^1 z\, B(z)\,dz = 1
\]
These kernel data explicitly determine both the stochastic breakage rate for log-sizes and the associated jump probability density.

## 2. Change of Variables and Mass-Weighted Log-Size Distribution

Transformation to log-size is given by
\[
s = \ln\frac{x}{x_0},\quad x = x_0 e^s,\quad dx = x_0 e^s ds
\]
Defining the mass-weighted (“tagged-mass”) log-size density
\[
p(s, t) = \frac{x^2 f(x, t)}{M} = \frac{x_0^2 e^{2s} f(x_0 e^s, t)}{M}
\]
with normalization $\int_{-\infty}^\infty p(s, t)\,ds = 1$, the master equation in log-size is derived by applying the kernel forms and change of variables to the mass flux.

## 3. Exact Log-Size Jump Generator and Pure-Jump Master Equation

The generator yields the following exact pure-jump master equation for $p(s, t)$:
\[
\boxed{
\partial_t p(s, t)
= -\lambda(s)p(s, t)
+ \int_0^\infty \lambda(s+u) K(u) p(s+u, t)\, du
}
\]
with log-size breakage rate and jump-law density given by
\[
\lambda(s) = k x_0^\alpha e^{\alpha s}
\]
\[
K(u) = e^{-2u} B(e^{-u}), \qquad u \geq 0
\]
and $\int_0^\infty K(u)\,du = 1$, ensured by the normalization of $B(z)$. In this process, the log-size $s$ remains constant between jumps; at rate $\lambda(s)$, it jumps to $s-u$, with $u$ sampled from $K(u)du$.

The backward (Kolmogorov) generator on test functions $f(s)$ is
\[
(Lf)(s) = \int_0^\infty \lambda(s)K(u) [f(s-u) - f(s)]\,du
\]
or, equivalently,
\[
Lf(s) = \int_{-\infty}^\infty ds' [W(s' \to s)f(s') - W(s \to s')f(s)]
\]
where the transition kernel is
\[
W(s \to s') = \lambda(s) K(s - s') \,\mathbf{1}_{s' \leq s}
\]
This structure is generically non-self-adjoint due to the lack of detailed balance between $\lambda(s)$ and $K(u)$.

## 4. Non-Hermitian Structure and Lindblad Embedding

The framework naturally lifts to a single-particle Hilbert space with orthonormal basis $\{|s\rangle\}$ and jump operators
\[
L(u) = \int_{-\infty}^{\infty} ds\, \sqrt{\lambda(s) K(u)} |s-u\rangle \langle s|
\]
The evolution of the density operator $\rho$ is governed by the Lindblad master equation:
\[
\boxed{
\partial_t \rho
= \int_0^\infty du \, [L(u)\rho L(u)^\dagger - \frac{1}{2}\{L(u)^\dagger L(u), \rho\}]
}
\]
When $\rho$ is diagonal in $|s\rangle$, the forward equation for $p(s, t)$ is exactly recovered. The absence of a Hermitian term means all non-unitarity is due to jumps; the effective non-Hermitian “no-jump” Hamiltonian is
\[
H_{\rm eff} = -\frac{i\hbar}{2} \int_0^\infty L(u)^\dagger L(u)\, du
\]
which controls relaxation, but is not itself the complete stochastic generator.

## 5. Second Quantization and Field-Theoretic Extensions

Generalization to NHQFT involves bosonic fields $a(s), a^\dagger(s)$ obeying $[a(s), a^\dagger(s')] = \delta(s - s')$. The Fock space jump operator is
\[
\hat{L}(u) = \int ds\, \sqrt{\lambda(s) K(u)}\, a^\dagger(s-u) a(s)
\]
The many-body Lindblad equation is:
\[
\partial_t \rho_F = \int_0^\infty du\, [\hat{L}(u) \rho_F \hat{L}(u)^\dagger - \frac{1}{2}\{\hat{L}(u)^\dagger \hat{L}(u), \rho_F\}]
\]
The one-body density $\langle \hat{n}(s) \rangle$ with $\hat{n}(s) = a^\dagger(s) a(s)$ obeys the exact log-size master equation; higher correlators are fixed by the Gaussian theory. In the Doi–Peliti path integral, shifting $a^\dagger \to 1 + \tilde{\varphi}$ leads to the action
\[
S_{\rm hop} = \int dt ds\, \tilde{\varphi}(s) \partial_t \varphi(s)
+ \int dt ds \int_0^\infty du\, \lambda(s) K(u)\, [\tilde{\varphi}(s) - \tilde{\varphi}(s-u)] \varphi(s)
\]
whose saddle-point conditions reproduce the master equation. The framework admits interacting generalizations, e.g., binary fragmentation with split-probability $\pi(z)$:
\[
\hat{L}_{\rm br}(z) = \int ds\, \sqrt{\lambda(s)\pi(z)}\, a^\dagger(s+\ln z) a^\dagger(s+\ln(1-z)) a(s)
\]
The corresponding Lindblad term introduces a genuine branching vertex, with mean-field reproducing the deterministic PBE and higher correlators capturing cascade noise.

## 6. Domain, Corner Conditions, and Universality

The log-size variable $s$ ranges over $\mathbb{R}$; practical boundaries (e.g., $s_{\rm min}$) may be imposed for removal of sub-detection sizes. $L$ acts on bounded, continuous, or $C^2$ functions decaying with $|s|\to\infty$. For jump-laws $K(u)$ sharply peaked near $u=0$, a Kramers–Moyal expansion is applicable, leading to a second-order approximation:
\[
\partial_t p \approx -\partial_s[v(s)p] + \partial_s^2 [D(s) p]
\]
with drift and diffusion coefficients $v(s) = -m_1\lambda(s)$, $D(s) = \frac{1}{2} m_2\lambda(s)$, $m_n = \int u^n K(u)\, du$. Upon similarity transformation, a non-Hermitian Schrödinger-type operator emerges, with a linear potential in the Airy sector enabling explicit mode-sum formulas for two-point correlations (see Sec. Airy of [2601.06635]).

Universality under coarse-graining is controlled by the inclusion of vertices: retention of only the Gaussian hopping theory yields linear transport universality; inclusion of branching interactions can lead to non-equilibrium critical points, e.g., directed-percolation-type fixed points in the presence of absorbing boundaries.

## 7. Summary and Significance

The Markov jump generator in log-size comprises a rigorous, exact stochastic framework for fragmentation processes, separating deterministic kernel data from stochastic modeling requirements. The connection to Lindblad dynamics, non-Hermitian generators, and NHQFT provides systematic access to fluctuations and correlations beyond mean-field PBEs. Explicit solutions and universality classification are available in quadratic (Airy) sectors and branching extensions, linking fragmentation kinetics with broader paradigms in stochastic processes and non-equilibrium field theory [2601.06635].

Source: https://www.emergentmind.com/topics/markov-jump-generator-in-log-size