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Biological Physics and Statistical Mechanics

Ensemble Theory of Active Fluctuations

Roman Belousov with J. Elliott and A. Erzberger

Erzberger Lab | EMBL, Heidelberg

Phys. Rev. E 113, 014133 (2026)

Politecnico di Torino (July, 2026)

Example: sorting by differential adhesion

Epi & PrE sorting — Epiblast and Primitive Endoderm in the inner cellular mass from a mouse blastocyst.

(CPM) — Poissonian cellular Potts models $E = \sum_{\text{voxels $i,j$ }} \gamma(s_i, s_j) + \sum_{\text{cells $c$}} \frac{\kappa_c}{2}(V_c - \bar{V}_c)^2$

Scheme of a CPM — $P_{dt}(x\pm dx|x) \propto \exp(-\beta \Delta E)$

Passive sorting

Simulations — Sorting with cell growth & division, and without.

Active sorting

Martian mouse — 😏?

Simulations — Effective temperature $T$ vs. active fluctuations $\phi$ $P_{dt}(x \pm dx|x) \propto \exp(-\beta \Delta E + \phi)$

Equilibrium ensembles

Microcanonical: MaxEnt, fixed $N, V, E$ , etc.
Canonical and other: Free energy, fixed temperature $T$ , etc.

Path ensembles

Caliber (path entropy): MaxCal with phenomenological constraints
Microcaliber: With microcanonical constraints

🤹‍♂️ Boltzmann gas

Realizations of a state (level occupancies $n^i$ with degeneracies ( $g^i$ ): $\ln \#(\text{state}) \approx \sum_{\text{levels}\ i} \left[g^i \ln n^i - n^i (\ln n^i - 1)\right]$

🤹‍♂️ Boltzmann gas in dynamics

Path realizations: transitions between levels $n^{ij}$ with multiplicities $\gamma^{ij}$ $\ln \#(\text{path}|dt) \approx \sum_{\text{from}\ j\ \text{to}\ i} \left[\gamma^{ij} \ln n^{ij} - n^{ij} (\ln n^{ij} - 1)\right]$

🏌️‍♂️ Ensemble theory of active fluctuations

Walkers on a lattice — $N$ walkers on a lattice ( $N_{\ell=1,2,...L}$ at site $\ell$ ): each walker exists in one of $M$ energy states

Microscopic constraints

Continuity of matter; Continuity of energy (general) or its conservation; Other constraints: Phys. Rev. E 113, 014133 (2026)
Traffic...

Traffic

Traffic — Traffic $\Phi$ vs. net current $J$ :

$J = \sum_{\ell > m} (N_{\ell m} - N_{m \ell}) = 0$ (opposing directions)

$\Phi = \sum_{\ell > m} (N_{\ell m} + N_{m \ell}) = 2$ cars

Lagrange multipliers

Thermodynamic $\beta = (k_{\rm B} T)^{-1}$
Traffic affinity $\eta$

⛳ Principle of maximum (micro-)caliber

CPM interface

Dynamics: $P(x\pm dx|x) \propto \exp(-\beta \Delta E + \phi)$ ,
where $\phi=-\eta$ .

Maximize path entropy

$N_\ell(t+dt) = \sum_m P_{\ell m}(\beta,\eta) N_m(t)$

Norton ensemble: fix energy $E$ and traffic $\Phi$
Thévenin ensemble: fix temperature $\beta$ and traffic affinity $\eta$

Enhanced diffusion

Simulations — Diffusion in a weak confinement $10^{-2} |\ell - 50|$ enhanced by $\eta$ .

Gillespie-like simulations with 10⁵ particles; solid/dashed curves show 451/4501 simulation steps.

Thank you for your attention!

Principled cat & Martian mouse

Questions? Comments?

Some illustrations are generated with ChatGPT and Gemini; prompt and selection by RB

Just an example...

Simulations — Prediction of mass and temperature gradient in hard-core gas between two thermalizing walls.

Boltzmann gas is calibrated from a single equilibrium simulation