Expression	Description	Units
Variables
$x[\tau ]=\{x(t):t\in (0,\tau )\}$	Trajectory or path through state space	a.u. (m)
$\omega (\tau )$	Random fluctuations	a.u. (m)
$x=\{\eta ,s,a,\mu \}\in X$	Markovian partition into external, sensory, active, and internal states	a.u. (m)
$\dot{x}=\frac{dx}{dt}$	Time derivative (Newton notation)	m/s
$\alpha =\{a,\mu \}\in A$	Autonomous states	a.u. (m)
$b=\{s,a\}\in B$	Blanket states	a.u. (m)
$\pi =\{b,\mu \}\in P$	Particular states	a.u. (m)
$\eta \in E$	External states	a.u. (m)
$\mathsf{\Gamma }={\mu }_m{k}_{B}T$	Amplitude (i.e., half the variance) of random fluctuations	J·s/kg
$Q$	Rate of solenoidal flow	J·s/kg
${\mu }_m=\frac{1}{k_{B}T}\mathsf{\Gamma }$	Mobility coefficient	s/kg
$T$	Temperature	K (Kelvin)
$\ell ={\displaystyle \int d\ell }:d{\ell }^2=g_{ij}d{\lambda }^{j}d{\lambda }^i$	Information length	nats
$\mathsf{\tau }:d\ell (\tau \ge \mathsf{\tau })\approx 0$	Critical time	s
$g_{ij}=E\left[\frac{\partial \mathfrak{I}}{\partial {\lambda }^i}\frac{\partial \mathfrak{I}}{\partial {\lambda }^j}\right]$	Fisher (information metric) tensor	a.u.
Functions, functionals and potentials
$f(x)$	The expected flow of states from any point in state space. This is the expected temporal derivative of x, averaging over random fluctuations in the motion of states.
$E[x]=E_p[x]={\displaystyle \int x{p}_{\lambda }(x)dx}$	Expectation or average
$p_{\lambda }(x):\mathrm{Pr}[X\in A]={\displaystyle {\int }_A{p}_{\lambda }(x)dx}$	Probability density function parameterised by sufficient statistics λ
$q_{\mu }(\eta )$	Variational density—an (approximate posterior) density over external states that is parameterised by internal states
$\mathcal{A}(x[\tau ])\equiv \mathfrak{I}(x[\tau ])$	Action: the surprisal of a path, i.e., the path integral of the Lagrangian
$U(\pi )=k_{B}T\cdot \mathfrak{I}(\pi )+\ln Z$	Thermodynamic potential	J or kg m2/s2
$F(\pi )\ge \mathfrak{I}(\pi )$	Variational free energy free energy—an upper bound on the surprisal of particular states	nats
$G(\alpha [\tau ])\ge \mathcal{A}(\alpha [\mathsf{\tau }]|{\pi }_0)$	Expected free energy free energy—an upper bound on the (classical) action of an autonomous path	nats
Operators
${\nabla }_x\mathfrak{I}(x)=\frac{\partial \mathfrak{I}}{\partial x}=\left(\frac{\partial \mathfrak{I}}{\partial x_1},\frac{\partial \mathfrak{I}}{\partial x_2},\dots \right)$	Differential or gradient operator (on a scalar field)
${\nabla }_{xx}\mathfrak{I}(x)=\frac{{\partial }^2\mathfrak{I}}{\partial x^2}$	Curvature operator (on a scalar field)
Entropies and potentials
$\mathfrak{I}(x)=-\ln p(x)$	Surprisal or self-information	nats
$D[q(x)||p(x)]=E_q[\ln q(x)-\ln p(x)]$	Relative entropy or Kullback–Leibler divergence	nats
(arbitrary units (a.u.), e.g., metres (m), radians (rad), etc.).