Any stable system linearized about a fixed point $x_\infty$ obeys $$\dot x = -M,(x - x_\infty).$$ Diagonalizing $M = \sum_i \lambda_i, u_i u_i^\top$, each mode decays as $e^{-\lambda_i t}$, so mode $i$ has relaxation time $\tau_i = 1/\lambda_i$. One state variable gives a single $\tau$; many give a spectrum of them. The “time constant” is therefore not a property of the wiring — it is an inverse eigenvalue of the operator that returns the system to equilibrium.

Instances

• RC circuit: $\tau,\dot V = -(V - V_\infty)$ with $\tau = RC$. A single eigenvalue $-1/\tau$.
• Drude model: $m,\dot v = -m v/\tau$, so $v(t) = v_0, e^{-t/\tau}$; $\tau$ is the electron scattering time and DC conductivity is $\sigma = n e^2 \tau / m$.
• Gradient descent: gradient flow $\dot x = -A x$ on a quadratic relaxes each Hessian eigendirection with $\tau_i = 1/\lambda_i$. The slowest mode sets convergence, and the condition number $\lambda_{\max}/\lambda_{\min}$ is the ratio of the longest to the shortest relaxation time.

The same structure recurs in Newton’s law of cooling, first-order chemical kinetics, the Ornstein–Uhlenbeck velocity process, and Debye/Maxwell relaxation — all single-eigenvalue (or single-mode) returns to equilibrium.

Caveat

The effective mass $m^{} = \hbar^2 / (d^2 E/dk^2)$ is not a relaxation time — it is a band curvature (an inertia, not a timescale). It is mentally adjacent to $\tau$ only because the two appear together in transport formulas such as the mobility $\mu = e\tau/m^{}$.