Expression	Description	Units
Variables
$\omega (\tau )$	Random fluctuations	a.u. (m)
$x=\{\eta ,s,a,\mu \}\in X$	Markovian partition into external, sensory, active, and internal states	a.u. (m)
$\alpha =\{a,\mu \}\in A$	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)
$\Gamma ={\mu }_m{k}_{B}T$	Amplitude (i.e., half the variance) of random fluctuations	J·s/kg
$Q$	Rate of solenoidal flow	J·s/kg
$\ell ={\displaystyle \int d\ell }:d{\ell }^2=g_{ij}d{\lambda }^{j}d{\lambda }^i$	Information length	nats
$g_{ij}=E\left[\frac{\partial \Im }{\partial {\lambda }^i}\frac{\partial \Im }{\partial {\lambda }^j}\right]$	Fisher (information metric) tensor	a.u.
Functions, functionals and potentials
$E[x]=E_p[x]={\displaystyle \int x{p}_{\lambda }(x)dx}$	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 $\lambda$
$q_{\mu }(\eta )$	Variational density – an (approximate posterior) density over external states that is parameterised by internal states
$F(b)\ge \Im (b)$	Variational free energy free energy – an upper bound on the surprisal of particular states	nats
Operators
${\nabla }_x\Im (x)=\frac{\partial \Im }{\partial x}=\left(\frac{\partial \Im }{\partial x_1},\frac{\partial \Im }{\partial x_2},\dots \right)$	Differential or gradient operator (on a scalar field)
Entropies and potentials
$\Im (x)=-\ln p(x)$	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