The Free Energy Landscape

The Free Energy Landscape

For systems amenable to such analysis, one can define an effective free energy functional:

where $U$ captures internal energy, $S$ entropy, and $T$ an effective temperature. The dynamics can often be written as:

for some positive-definite mobility tensor $\Gamma$. In this representation:

• Local minima of $\mathcal{F}$ correspond to metastable attractors
• Saddle points determine transition rates between attractors
• The depth of minima relative to barriers determines persistence times

One structure within this landscape will recur throughout the book. For a self-maintaining system, the viability manifold $\viable \subset \R^n$ is the region of state space within which the system can persist indefinitely (or for times long relative to observation scales):

where $\tau_{\text{exit}}(\mathbf{x})$ is the first passage time to a dissolution state starting from $\mathbf{x}$.

The viability manifold will play a central role in understanding normativity: trajectories that remain within $\viable$ are, in a precise sense, “good” for the system, while trajectories that approach the boundary $\partial\viable$ are “bad.”