The Langevin Equation

The Langevin equation, introduced by Paul Langevin in 1908, is the foundational dynamical description of Brownian motion. Rather than tracking the astronomically many collisions a pollen grain suffers from solvent molecules, Langevin split the total force on the particle into two physically distinct pieces: a slowly varying systematic part — the viscous drag $-\gamma v$ that a macroscopic object feels moving through a fluid — and a rapidly varying random part $\xi(t)$ that averages to zero but whose fluctuations actually drive the diffusive wandering. The decomposition is the conceptual heart of stochastic dynamics, and Langevin himself called it “infinitely simpler” than Einstein's 1905 diffusion argument. The deep self-consistency of the theory is the fluctuation-dissipation principle: the same microscopic interactions that produce momentum dissipation through friction $\gamma$ also generate the random force, with its strength fixed by the same thermal environment. They cannot be chosen independently; thermal equilibrium at temperature $T$ forces the noise correlation to be $\langle\xi(t)\xi(t')\rangle = 2\gamma k_B T\,\delta(t-t')$ . Were this not so, the particle would either heat up or cool relative to its surroundings, violating the second law. This single constraint reproduces equipartition $\langle v^2\rangle=k_BT/m$ for the velocity and the Einstein relation $D=k_BT/\gamma$ for diffusion simultaneously. Mathematically the Langevin equation is a stochastic differential equation. The white-noise force is the formal derivative of the Wiener process, an object so singular that ordinary calculus must be replaced by stochastic calculus, with the Itô and Stratonovich conventions giving different answers when the noise multiplies a state-dependent function. The probabilistic content — the time-dependent distribution of velocity and position — is captured equivalently by the Fokker–Planck equation. Because it embeds dissipation, fluctuations, and a deterministic force law in one compact equation, the Langevin framework generalizes far beyond a single colloidal bead: to noise-activated barrier crossing, to colored noise and memory kernels, and to the modern theory of linear response, where the relaxation of spontaneous fluctuations governs the response to external perturbations.

Key takeaways

The Langevin equation models a Brownian particle as obeying Newton's law with two extra forces: a systematic viscous drag $-\gamma v$ and a rapidly fluctuating random force $\xi(t)$ representing molecular collisions.
The random force is idealized as Gaussian white noise with zero mean and a delta-correlated autocorrelation fixed by fluctuation-dissipation: $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$ .
Velocity relaxes on the timescale $\tau=m/\gamma$ toward thermal equilibrium, where $\langle v^2\rangle=k_BT/m$ by equipartition; at long times, the position diffuses with $D=k_BT/\gamma$ , the Einstein relation.
The velocity autocorrelation decays exponentially, $\langle v(t)v(0)\rangle=(k_BT/m)e^{-|t|/\tau}$ , and its time integral yields the diffusion constant $D$ .
The equation is the prototype stochastic differential equation; its probability density obeys the Fokker–Planck equation, while the distinction between Itô and Stratonovich interpretations becomes essential when the noise is multiplicative.
Key idea: replace an intractable many-body force by a mean drag plus a random force, and demand consistency with thermal equilibrium — this single requirement (fluctuation–dissipation) ties the noise strength $2\gamma k_BT$ to the friction and yields both equipartition and the Einstein relation.

The core insight

Replace an intractable many-body force by a mean drag plus a random force, and demand consistency with thermal equilibrium — this single requirement (fluctuation–dissipation) ties the noise strength to the friction and yields both equipartition and the Einstein relation.

Where this lives today

The Langevin equation is a working tool across modern physics, chemistry, and biology. In molecular-dynamics simulation it is implemented as a stochastic thermostat: adding $-\gamma v + \xi$ to each atom's equation of motion drives the system to the canonical ensemble at temperature $T$ , and the fluctuation–dissipation balance guarantees correct equilibrium sampling regardless of the chosen $\gamma$ . In single-molecule biophysics, optical-tweezer and video-tracking experiments fit measured mean-square displacements to the Ornstein–Fürth form to extract local viscosities, applied forces, or the elasticity of DNA and molecular motors. The same mathematics reappears far from colloids. Johnson–Nyquist thermal noise in a resistor provides an electrical analogue of Langevin fluctuations, with voltage-noise spectral density $S_V(f)=4k_BT R$ — the electrical incarnation of fluctuation–dissipation. Generalizations with memory kernels and colored noise (the generalized Langevin equation) underlie the modern theory of relaxation in glasses and proteins, and the barrier-crossing version is the foundation of chemical reaction-rate theory, developed further in Kramers' Escape Rate Theory.

The physical picture

Picture a heavy bead jostled by a swarm of tiny solvent molecules. On the bead's slow timescale the collisions look like two effects: when it moves, the molecules in front hit harder than those behind, producing a net drag $-\gamma v$ ; and even when it sits still, the random imbalance of hits gives a fluctuating kick $\xi(t)$ . The drag drains energy; the kicks supply it. Equilibrium is the steady state where the two exactly balance. This is the phenomenon of Brownian motion, whose observed mean-square displacement $\langle\Delta x^2\rangle\to 2Dt$ the Langevin equation must reproduce from a dynamical law. The separation of timescales is essential: molecular collisions happen on roughly $10^{-13}\,\mathrm{s}$ , while the bead's velocity relaxes over $\tau=m/\gamma$ , which for a micron-sized bead in water is typically of order $10^{-7}\,\mathrm{s}$ . Because the noise correlation time is vastly shorter than $\tau$ , it is legitimate on the bead's dynamical timescale to idealize $\xi(t)$ as delta-correlated white noise — the bead cannot resolve the detailed structure of individual molecular kicks. Friction and fluctuation are two faces of one coin. A larger $\gamma$ means more collisions, hence stronger drag and stronger noise. The fluctuation–dissipation theorem encodes exactly this: you may not dial the noise down without also reducing the friction, on pain of violating the second law. The punchline figure below makes the constraint visible: the velocity correlation decays like $e^{-t/\tau}$ , but its value at $t=0$ is pinned to $k_BT/m$ — precisely the amplitude the noise strength $2\gamma k_BT$ was tuned to produce.

Figure

Velocity autocorrelation decays exponentially, with its

t = 0

value pinned by equipartition to

k_B T/m

— the fluctuation-dissipation constraint made visible

Mathematically, the deterministic part pulls the velocity toward zero exponentially, while the noise constantly re-randomizes it. The competition produces a stationary Gaussian velocity distribution — the Maxwell–Boltzmann form — independent of initial conditions. Position, lacking any restoring force, never reaches a stationary state: it diffuses forever, crossing over from an inertial $t^2$ regime at short times to the diffusive $2Dt$ law once velocity has decorrelated.

Figure

Force decomposition: the systematic drag −γv opposes motion while the random kicks ξ(t) re-randomize it, and their balance sets the thermal steady state

The core equations

The Langevin equation replaces Newton's second law for a particle in a fluid by adding, to the systematic viscous drag, a fluctuating force that stands in for the countless individual molecular collisions. For a free particle (no external potential) this reads as follows.

m\,\frac{\mathrm{d}v}{\mathrm{d}t} = -\gamma\,v + \xi(t) \qquad (1)

where: $m$ — mass of the Brownian particle; $v$ — particle velocity; $\gamma$ — friction (drag) coefficient, for a sphere of radius $a$ in a fluid of viscosity $\eta$ given by Stokes' law $\gamma=6\pi\eta a$ ; $\xi(t)$ — random fluctuating force, idealized as Gaussian white noise. Physically: the drag term systematically slows the particle, the noise term randomly re-accelerates it. The random force cannot be an arbitrary noise. To keep the particle in thermal equilibrium with the fluid, its strength is locked to the friction by the fluctuation–dissipation relation, which we derive below.

\langle \xi(t)\rangle = 0,\qquad \langle \xi(t)\,\xi(t')\rangle = 2\gamma k_B T\,\delta(t-t') \qquad (2)

where: $k_B$ — Boltzmann constant; $T$ — temperature of the surrounding fluid; $\delta(t-t')$ — Dirac delta encoding a vanishing noise correlation time (white noise). The zero mean says the random force has no systematic bias; the delta correlation idealizes successive kicks as uncorrelated on the resolved timescale; and the coefficient $2\gamma k_BT$ is fixed by the fluctuation–dissipation relation. Once the noise strength is fixed consistently with the friction, the velocity relaxes to a stationary thermal state whose correlations decay on the single timescale $\tau=m/\gamma$ . Integrating the equation gives the velocity autocorrelation and the momentum relaxation time.

\tau = \frac{m}{\gamma},\qquad \langle v(t)\,v(0)\rangle = \frac{k_B T}{m}\,e^{-|t|/\tau} \qquad (3)

where: $\tau$ — velocity (momentum) relaxation time; $\langle v(t)v(0)\rangle$ — stationary velocity autocorrelation function. The $t=0$ value $k_BT/m$ is fixed by equipartition; the decay rate $1/\tau$ is set by the drag. The diffusion coefficient is not an independent parameter either: it is the running time-integral of the velocity correlation, a relation that recovers the Einstein value.

D = \int_0^{\infty}\!\langle v(t)\,v(0)\rangle\,\mathrm{d}t = \frac{k_B T}{\gamma} \qquad (4)

where: $D$ is the diffusion coefficient of the particle's position. This is the seed of the general fluctuation–dissipation theorem: a transport coefficient such as $D$ can be expressed as an integral of an equilibrium time-correlation function. The full mean-square displacement interpolates between the two regimes visible in the physical picture, and is called the Ornstein–Fürth formula.

\langle \Delta x^2(t)\rangle = \frac{2k_BT}{\gamma}\left[t - \tau\left(1-e^{-t/\tau}\right)\right] \qquad (5)

where: $\Delta x(t)=x(t)-x(0)$ . At short times ( $t\ll\tau$ ) the bracket $\to t^2/(2\tau)$ giving ballistic $\langle\Delta x^2\rangle\to (k_BT/m)t^2$ ; at long times ( $t\gg\tau$ ) the bracket $\to t$ giving diffusive $2Dt$ . Adding an external potential turns the free equation into the equation used for reaction-rate and barrier-crossing problems.

m\,\frac{\mathrm{d}v}{\mathrm{d}t} = -\gamma\,v - \frac{\mathrm{d}U}{\mathrm{d}x} + \xi(t) \qquad (6)

where: $U(x)$ — external potential energy, with conservative force $-U'(x)$ . The stochastic dynamics of this equation is described equivalently by the Fokker–Planck equation, and in the inertia-free limit by the overdamped (Smoluchowski) equation $\gamma\dot x = -U'(x)+\eta(t)$ .

Derivation ASolving the free Langevin equation: relaxation, equipartition, and diffusion

This derivation integrates the linear stochastic equation of motion for the velocity, computes the stationary variance and the autocorrelation function, and then obtains the mean-square displacement and the Einstein relation. We treat $m\dot v = -\gamma v+\xi$ as a linear first-order equation with a stochastic source, using only the noise statistics of equation (2). The goal is to show that a single fluctuation–dissipation input — the noise strength $2\gamma k_BT$ — together with the friction $\gamma$ produces equipartition, exponential velocity correlations, and normal diffusion.

Step 1. Solve the linear equation by variation of parameters with $\tau=m/\gamma$ :

v(t) = v(0)e^{-t/\tau} + \frac{1}{m}\int_0^t e^{-(t-t')/\tau}\,\xi(t')\,\mathrm{d}t'.

Why: the homogeneous part relaxes deterministically; the particular part accumulates the random kicks weighted by how recently they arrived.

Step 2. Average using $\langle\xi\rangle=0$ : $\langle v(t)\rangle = v(0)\,e^{-t/\tau}$ . Why: the noise contributes nothing to the mean, so the average velocity simply decays.

Step 3. Square and average, inserting $\langle\xi(t')\xi(t'')\rangle=2\gamma k_BT\,\delta(t'-t'')$ :

\langle v^2(t)\rangle = v(0)^2 e^{-2t/\tau} + \frac{2\gamma k_B T}{m^2}\int_0^t e^{-2(t-t')/\tau}\,\mathrm{d}t'.

Why: the delta function collapses the double time integral to a single one.

Step 4. Evaluate the integral: $\langle v^2(t)\rangle=v(0)^2e^{-2t/\tau}+\frac{k_BT}{m}(1-e^{-2t/\tau})$ . As $t\to\infty$ , the memory of the initial velocity is lost and $\langle v^2\rangle\to k_BT/m$ , recovering equipartition. This is exactly what fixing the noise strength to $2\gamma k_BT$ delivers.

Step 5. In the stationary state the velocity autocorrelation is $\langle v(t)v(0)\rangle = (k_BT/m)\,e^{-|t|/\tau}$ , decaying on the momentum time $\tau$ .

Step 6. Time-integrate the autocorrelation: $D=\int_0^\infty (k_BT/m)e^{-t/\tau}\,\mathrm{d}t=(k_BT/m)\tau=k_BT/\gamma$ . Why: this Green–Kubo integral gives the Einstein relation directly, recovering the diffusion coefficient that Brownian Motion observes as $\langle\Delta x^2\rangle=2Dt$ . These four results are mutually consistent: the stationary variance forces $\Gamma=2\gamma k_BT$ ; the autocorrelation intercept equals $k_BT/m$ ; the Green–Kubo integral gives $D=k_BT/\gamma$ ; and the short- and long-time MSD limits reproduce $t^2$ and $2Dt$ respectively.

Go deeper · Derivation B · 1 minFluctuation–dissipation: why the noise strength is fixed by the friction

Derivation BFluctuation–dissipation: why the noise strength is fixed by the friction

This derivation shows that thermal equilibrium uniquely determines the noise variance, tying it to the dissipative coefficient $\gamma$ . Rather than solving the full dynamics, we impose the single physical requirement that the average kinetic energy be stationary at the thermal value and read off the constraint on the noise. This is the fluctuation–dissipation relation of the second kind, and it is the reason the Langevin equation is thermodynamically consistent rather than an arbitrary damped-driven oscillator.

Step 1. Multiply the Langevin equation by $v$ and average to get the energy balance:

\frac{\mathrm{d}}{\mathrm{d}t}\left\langle \tfrac12 m v^2\right\rangle = -\frac{\gamma}{m}\langle mv^2\rangle + \langle v\,\xi\rangle.

Why: dissipation ( $-\gamma$ ) cools the particle, the noise correlation heats it; equilibrium is where they cancel.

Step 2. Compute the noise–velocity correlation from the formal solution with $\langle\xi\xi'\rangle=\Gamma\delta$ :

\langle v(t)\,\xi(t)\rangle = \frac{1}{m}\int_0^t e^{-(t-t')/\tau}\Gamma\,\delta(t-t')\,\mathrm{d}t' = \frac{\Gamma}{2m}.

Why: with the usual symmetric convention for the Dirac delta at an integration endpoint, $\int_0^t\delta(t-t')f(t')\,dt'=\tfrac12 f(t)$ , giving the factor $1/2$ .

Step 3. Impose the steady state $\mathrm{d}\langle\tfrac12 mv^2\rangle/\mathrm{d}t=0$ : $0=-\dfrac{2\gamma}{m}\langle\tfrac12 mv^2\rangle+\dfrac{\Gamma}{2m}$ .

Step 4. Insert equipartition $\langle\tfrac12 mv^2\rangle=\tfrac12 k_BT$ and solve: $\Gamma = 2\gamma k_BT$ . Why: this is the only noise strength consistent with the second law; a smaller value would let the particle cool below the bath, a larger one would heat it above. The energy-balance steady-state condition returns $\Gamma=2\gamma k_BT$ , identical to the value obtained from the explicit solution in the derivation above.

Limiting cases

The behavior splits cleanly across timescales and parameter regimes:

Case	Result	Meaning
Ballistic (short time, $t\ll\tau$ )	$\langle \Delta x^2(t)\rangle \simeq \frac{k_B T}{m}\,t^2$	Before friction acts, the particle moves inertially at its thermal speed; the mean-square displacement grows as $t^2$ .
Diffusive (long time, $t\gg\tau$ )	$\langle \Delta x^2(t)\rangle \simeq 2D\,t,\ \ D=\frac{k_B T}{\gamma}$	Velocity has decorrelated; the position performs a random walk and the Einstein relation appears.
Overdamped limit ( $m\to 0$ or $\gamma$ large)	$\gamma\,\dot{x} = F(x) + \eta(t)$	Inertia negligible; the equation becomes first-order (Smoluchowski). The regime of most colloidal and biophysical systems.
Stationary velocity distribution	$P_{\mathrm{st}}(v)\propto \exp\!\left(-\frac{m v^2}{2 k_B T}\right)$	Independent of initial condition; equipartition $\langle v^2\rangle=k_BT/m$ holds once relaxed.

Common trapWhite noise is singular, and the factor of 2 is not optional

White noise is not an ordinary function: $\xi(t)$ has infinite variance and $\langle\xi(t)\xi(t')\rangle\propto\delta(t-t')$ . Treating it naively breaks down when the noise is multiplicative — then the result depends on the Itô vs Stratonovich interpretation, discussed in Stochastic Calculus: Itô versus Stratonovich. Also, the coefficient $2\gamma k_BT$ (not $\gamma k_BT$ ) is mandatory: the factor of 2 is fixed by requiring the stationary Langevin dynamics to satisfy equipartition, $\langle v^2\rangle=k_BT/m$ . In a one-sided integral, the symmetric endpoint convention contributes an additional factor of $1/2$ , which is consistent with this result. Dropping the factor of 2 would give the wrong equilibrium temperature.

Worked examples

Example 1 — Diffusion coefficient and relaxation time of a micron bead (self-consistency with Brownian Motion).

Step 1. Take a sphere of radius $a=1\,\mu\mathrm{m}$ in water at $T=300\,\mathrm{K}$ , viscosity $\eta=10^{-3}\,\mathrm{Pa\,s}$ . Stokes' law gives the friction coefficient $\gamma=6\pi\eta a = 6\pi(10^{-3})(10^{-6}) = 1.885\times10^{-8}\,\mathrm{kg/s}$ .

Step 2. The Einstein relation from the Green–Kubo integral gives $D=k_BT/\gamma = (1.380649\times10^{-23})(300)/(1.885\times10^{-8})$ , so $D=\mathbf{2.197\times10^{-13}\ m^2/s}$ . This is the identical number Brownian Motion and The Einstein Relation obtain from the Stokes–Einstein form $k_BT/(6\pi\eta a)$ — a three-article closure.

Step 3. The momentum relaxation time uses the bead mass $m=\tfrac43\pi a^3\rho\approx 4.19\times10^{-15}\,\mathrm{kg}$ (water density). Then $\tau=m/\gamma = 4.19\times10^{-15}/1.885\times10^{-8} = \mathbf{2.22\times10^{-7}\ s}$ . Since any microscope observation time far exceeds $\tau$ , the measured motion is purely diffusive — the ballistic regime is hidden below $\sim0.2\,\mu\mathrm{s}$ .

Example 2 — The crossover mean-square displacement at $t=\tau$ .

Step 1. Evaluate the Ornstein–Fürth formula (5) at $t=\tau$ : $\langle\Delta x^2(\tau)\rangle = (2k_BT/\gamma)\,[\tau-\tau(1-e^{-1})] = (2k_BT/\gamma)\,\tau e^{-1} = 2D\tau e^{-1}$ .

Step 2. Numerically $2D\tau = 2(2.197\times10^{-13})(2.22\times10^{-7}) = 9.76\times10^{-20}\,\mathrm{m^2}$ , so $\langle\Delta x^2(\tau)\rangle = 9.76\times10^{-20}\times e^{-1} = \mathbf{3.59\times10^{-20}\ m^2}$ , an RMS displacement of about $1.9\times10^{-10}\,\mathrm{m}$ . The particle has barely moved an ångström-scale distance during one relaxation time — confirming that all observable Brownian wandering happens deep in the diffusive regime.

Origins · 1 minPaul Langevin and how this was discovered

Paul Langevin

1872–1946 · French physicist; student of Pierre Curie, professor at the Collège de France, and a leading figure in early relativity and magnetism

Paul Langevin reformulated Einstein's 1905 theory of Brownian motion in a single 1908 note to the Comptes Rendus, writing down the equation that now bears his name and calling his derivation "infinitely simpler" than Einstein's. By splitting the force on a suspended particle into a mean drag and a fluctuating remainder, he founded the language of stochastic dynamics. Langevin also developed the classical theory of paramagnetism and diamagnetism, pioneered ultrasonic echo-ranging (an ancestor of sonar) during the First World War, and was an early champion of Einstein's relativity in France — the relativistic "twin" thought experiment is often attributed to his 1911 lecture. A committed anti-fascist, he was briefly imprisoned under the Vichy regime.

History

1827

— Robert Brown reports the ceaseless jiggling of suspended particles

1905

— Einstein derives the diffusion theory of Brownian motion, ⟨x²⟩ = 2Dt

1906

— Smoluchowski publishes an independent kinetic derivation

1908

— Langevin writes m v̇ = −γv + ξ(t), the stochastic equation of motion

1908

— Perrin begins experiments confirming the theory and measuring N_A

1926

— Perrin awarded the Nobel Prize for the discontinuous structure of matter

1930

— Ornstein and Uhlenbeck give the rigorous theory of the velocity process

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 (Fluctuations and Brownian Motion) · Kardar, Statistical Physics of Particles, Ch. 9 · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 15 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 10 · Landau & Lifshitz, Statistical Physics Part 1, §118 · Risken, The Fokker-Planck Equation · Langevin, P. (1908), Comptes Rendus 146, 530 · van Kampen, Stochastic Processes in Physics and Chemistry. The key self-consistency checks are: the free-Langevin solution gives $\langle v^2\rangle_{\mathrm{st}}=k_BT/m$ only when $\Gamma=2\gamma k_BT$ ; the velocity autocorrelation $(k_BT/m)e^{-|t|/\tau}$ and the Green–Kubo integral together give $D=k_BT/\gamma$ ; the Ornstein–Fürth MSD correctly interpolates between the $t^2$ and $2Dt$ limits; and Stokes drag $\gamma=6\pi\eta a$ yields the numerical values $D=2.197\times10^{-13}\,\mathrm{m^2/s}$ and $\tau=2.22\times10^{-7}\,\mathrm{s}$ for the worked bead.

Builds on: Brownian Motion · Wiener Process and White Noise · Equipartition Theorem. Leads to: The Ornstein-Uhlenbeck Process · The Fokker-Planck Equation · The Einstein Relation · The Fluctuation-Dissipation Theorem · Kramers' Escape Rate Theory.

Is the Langevin equation just Newton's law with a fudge added?

Not a fudge — a systematic coarse-graining. Newton's law for the bead plus every solvent molecule is exact but intractable. Langevin's move is to integrate out the fast solvent degrees of freedom, which leaves two residues: their mean effect (the drag $-\gamma v$ ) and their fluctuating residue (the noise $\xi$ ). The Mori–Zwanzig projection formalism makes this rigorous and shows the drag and noise are not independent — they are the systematic and fluctuating parts of the same eliminated forces, which is precisely why fluctuation–dissipation locks them together.

Where does the factor of 2 in ⟨ξξ'⟩ = 2γk_BT δ actually come from?

It is forced by consistency, not chosen. Solve the dynamics with an unknown strength $\Gamma$ : the stationary velocity variance comes out to $\Gamma/(2m\gamma)$ . Equipartition demands this equal $k_BT/m$ , which gives $\Gamma=2\gamma k_BT$ . The apparent "2" reflects that a delta function at the endpoint of a one-sided time integral contributes only half its weight, so the noise variance must be twice as large to compensate. Any other coefficient would drive the particle's temperature away from the bath's.

Why is the velocity autocorrelation exponential and not something more complicated?

Because the free Langevin equation is linear and the noise is white (memoryless). A linear first-order equation driven by delta-correlated noise is an Ornstein–Uhlenbeck process, whose stationary correlation function is a single decaying exponential $e^{-t/\tau}$ with $\tau=m/\gamma$ . Non-exponential (e.g. power-law or oscillatory) correlations appear only when you add memory friction (colored noise, the generalized Langevin equation) or a confining potential, both of which break the simple linear-white-noise structure.

How is the Langevin equation related to the Fokker–Planck equation?

They are two descriptions of the same stochastic process. The Langevin equation follows individual trajectories $v(t)$ ; the Fokker–Planck equation follows the probability density $P(v,t)$ over an ensemble of them. Because Gaussian white noise has only first and second moments, the Kramers–Moyal expansion truncates exactly at second order, converting the Langevin dynamics into a drift-plus-diffusion partial differential equation for $P$ . Averages computed either way agree; you choose whichever is more convenient. See The Fokker-Planck Equation.

Does the ballistic t² regime ever matter experimentally?

For a micron colloidal bead in water it is essentially invisible: the crossover time $\tau=m/\gamma$ is $\sim0.2\,\mu\mathrm{s}$ , far shorter than any optical measurement. But it is real and has been observed. High-bandwidth optical-trap experiments on trapped beads and on particles in air (where $\tau$ is much longer, because the friction is smaller) have directly resolved the inertial $\langle\Delta x^2\rangle\propto t^2$ regime and the smooth crossover to diffusion, confirming the Ornstein–Fürth formula. The velocity of a Brownian particle is a genuine, if fleeting, physical quantity.

Can I always use the overdamped (Smoluchowski) equation and drop inertia?

For most soft-matter and biological problems, yes: on the timescales of interest the velocity has already relaxed, so $\gamma\dot x = -U'(x)+\eta(t)$ is an excellent approximation and vastly simpler. The overdamped limit is valid whenever the observation time greatly exceeds $\tau=m/\gamma$ . It fails when inertia matters — very short times, low-friction environments (dilute gases, levitated nanoparticles), or resonant systems — where the full second-order equation with the $m\ddot x$ term is required.