Treatise 12: Kramers Barrier Escape Model & Topological Transition Rates

J. McKenney

This is a short-form treatise in the Mathematical Physics Models working group, a companion note to MP-MATH-03 (Sheaf Cohomology & Topological Fault Localization in Cyber-Physical Distribution Graphs), which cites this model directly for its physical fault-barrier framing, and its escape-rate coefficient is cataloged as formula C8 in the group's consolidated CDT Mathematical Models reference. It is a self-contained physics analogy rather than a full treatment, and a reader does not need either sibling document to follow it.

Licence: CC BY 4.0. 17 September 2026.

Executive Abstract#

This note models the probability that a monitored system moves from a secure state to a compromised one by borrowing a result from the physical chemistry of reaction rates rather than building a probability model from scratch. The borrowed result, Kramers' theory of escape over an energy barrier, describes how a particle trapped in a potential well occasionally gathers enough energy to escape; the note treats a defended network the same way, as a particle held in a secure well by a barrier built from segmentation, detection coverage, and isolation.

The payoff is a small set of quantities that map onto what a security team already measures: a barrier height that falls as segmentation weakens, an attempt frequency that rises with how densely an attacker's neighborhood in the network graph is connected, and a temperature term standing in for attacker sophistication. Combined through the same exponential form used in the original chemistry, they yield an escape rate and its inverse, a mean time to compromise, for a given actor against a given segment.

The Seldon platform uses the resulting rate three ways: to rank actors by how readily they tunnel through the barriers they face, to flag unusually thin barriers for hardening, and to bias Monte Carlo walk simulations so segments with a higher computed escape rate are reached more often.

Abstract#

In the Seldon Cyber Digital Twin (CDT), we adapt Kramers' Transition State Theory from physical chemistry to cybersecurity topology. A system, such as a power grid control network, is a particle trapped in a potential well, the Secure state; a breach requires the adversary to escape the well by overcoming a barrier of height Delta E. The escape rate k, the frequency of successful transitions per unit time, follows the Arrhenius-like equation of Section 1, with barrier height set by shortest path distance and edge weight sum in the Neo4j/pgvector graph, attempt frequency by neighborhood connectivity density, and a Threat Temperature term k_B T for attack sophistication. Its inverse gives a mean time to compromise for a given actor against a given segment.

1. Theoretical Foundation#

In the Seldon Cyber Digital Twin (CDT), we adapt Kramers' Transition State Theory from physical chemistry to cybersecurity topology. In this model, a system (e.g., a power grid control network) is viewed as a particle trapped in a potential well (the "Secure" state). For an adversary to achieve a breach, they must "escape" this well by overcoming a potential barrier ($\Delta E$).

The Escape Rate Formula#

The escape rate $k$, representing the frequency of successful transitions (breaches) per unit time, is defined by the Arrhenius-like equation:

Where:

• $k$: The transition probability per unit time (Escape Rate).
• $A$: The pre-exponential factor (Collision frequency/Attempt frequency), modeled as the connectivity density of the actor's neighborhood in the graph.
• $\Delta E$: The Barrier Height, representing the topological resistance (defensive posture, isolation, air-gaps).
• $k_B T$: The Threat Temperature, a stochastic noise term representing the Attack Sophistication (e.g., APT = 1.5, Nation-State = 2.0).

2. CDT Implementation#

Topological Mapping#

In our Neo4j/pgvector graph, $\Delta E$ is calculated as a function of the shortest path distance and edge weight sum between the Actor node and the Target node.

• High $\Delta E$: Strong network segmentation, EDR coverage, and restricted lateral movement edges.
• Low $\Delta E$: Flat networks, exposed credentials, and high edge density.

Mean Time to Compromise (MTTC)#

The MTTC is the inverse of the escape rate:

This metric provides a temporal forecast of how long a specific actor (given their $k_B T$) will take to breach a specific segment.

3. Application in Seldon Intelligence#

Seldon Intelligence consumes the escape rate and mean time to compromise in three ways:

1. Actor Ranking: Actors are ranked by their ability to "tunnel" through high barriers (High $k_B T$).
2. Topological Hardening: Seldon identifies "thin" barriers where $\Delta E$ is critically low and recommends edge deletions (e.g., "Delete cross-segment service account") to increase the barrier height.
3. Monte Carlo Validation: Our Monte Carlo walks are "biased" by these escape rates; higher $k$ values increase the transition probability of a walk moving from a source node to a target node.

4. References#

• [1] Kramers, H. A. (1940): "Brownian motion in a field of force and the diffusion model of chemical reactions." Physica.
• [2] Hanggi, P., et al. (1990): "Reaction-rate theory: fifty years after Kramers." Reviews of Modern Physics.
• [3] McKenney, J. (2025): Topological Cyber-Physics: Foundations of the Digital Twin. Eigenia Labs working paper.