Log-Size Markov Jump Generator

• The paper introduces an exact stochastic operator that models particle fragmentation via jumps in log-size, derived from deterministic PBEs.
• It employs a change of variables and Lindblad embedding to transition from deterministic breakage rates to a non-self-adjoint quantum framework.
• The method extends to non-Hermitian quantum field theory, enabling analysis of size correlations, fluctuations, and non-equilibrium universality in fragmentation.

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 (Segura, 10 Jan 2026).

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 (Segura, 10 Jan 2026)).

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 (Segura, 10 Jan 2026).