Onsager Regression Hypothesis

Watch the height of a fluid column in a thin tube. Random molecular impacts make it jitter above and below its mean. Pick a moment when, by chance, it sits too high — and ask how it comes back down. Lars Onsager's 1931 hypothesis gives a startling answer: on average, that spontaneous excess drains away following exactly the same equation you would write for a column you had deliberately filled too high with a pipette. The system carries no memory of the cause of its deviation, only of the deviation itself. A fluctuation and an experimenter's push, once released, are statistically indistinguishable in their subsequent decay. The power of this statement is that it welds together two quantities measured in completely different ways. On one side is the equilibrium time-correlation function $\langle\delta A(0)\delta A(t)\rangle$ , an intrinsic property of the undisturbed, fluctuating system — the kind of thing you extract from an equilibrium molecular-dynamics run. On the other side is a macroscopic transport coefficient — a diffusion constant, a conductivity, a viscosity — defined operationally by the response to an imposed gradient. Onsager regression says these are two faces of one object, and it is the physical content that becomes the Green–Kubo formulas and the classical fluctuation–dissipation theorem. This article proves the hypothesis from linear response and microscopic time-reversal symmetry, shows it is exact for Gaussian–Markov dynamics using the velocity autocorrelation of a Brownian particle as the clean exhibit, and identifies regression as the classical fluctuation–dissipation theorem written in the time domain. It is the conceptual engine that lets equilibrium simulations predict transport, and the second pillar (alongside correlation symmetry) beneath the Onsager reciprocal relations.

Key takeaways

The regression hypothesis (Onsager 1931): the average decay of a spontaneous equilibrium fluctuation is governed by the same macroscopic laws that describe the relaxation of a deliberately imposed deviation of the same observable.
Formally, the normalized equilibrium autocorrelation $C(t)=\langle\delta A(0)\delta A(t)\rangle/\langle\delta A^2\rangle$ obeys the same linear relaxation equation as the macroscopic variable $\overline{\Delta A}(t)$ .
It follows from linear-response theory plus microscopic time-reversal symmetry; it is a theorem, not a separate postulate, once linear response is granted.
It converts microscopic correlation functions into measurable transport coefficients — the Green–Kubo relation $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt$ is its direct consequence.
It is exact for Gaussian–Markov (linear-Langevin) dynamics; the velocity autocorrelation $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma t}$ decays at the very rate $\gamma$ that damps a macroscopic drift.
Read modernly, regression is the classical fluctuation–dissipation theorem in the time domain: $\chi(\tau)=-\beta\,\Theta(\tau)\,\dot C(\tau)$ .
Key idea: a spontaneous fluctuation and an externally prepared deviation of the same observable are statistically indistinguishable in their subsequent average decay; hence the equilibrium autocorrelation function and the macroscopic relaxation law are the same function of time.

Where this lives today

Onsager's regression hypothesis underlies the modern computation of transport coefficients from equilibrium molecular-dynamics simulations. In equilibrium molecular dynamics, no external driving field is applied. Instead, one computes equilibrium autocorrelation functions of the velocity, stress tensor, or heat current and evaluates the corresponding Green–Kubo integrals to obtain diffusion coefficients, viscosities, and thermal conductivities. The hypothesis, together with linear-response theory, justifies replacing macroscopic transport coefficients by equilibrium time-correlation integrals. This is why a single equilibrium simulation can predict many transport coefficients without explicitly driving the system out of equilibrium. In materials discovery pipelines and in first-principles molecular dynamics, this is the standard route to transport properties. The same principle appears throughout spectroscopy and scattering. Dielectric and magnetic susceptibilities $\chi(\omega)$ are Fourier transforms of equilibrium polarization or magnetization autocorrelations; light- and neutron-scattering line shapes (the dynamic structure factor $S(\mathbf q,\omega)$ ) are Fourier transforms of density–density correlations, whose small- $\mathbf q$ decay follows the macroscopic diffusion equation with Rayleigh linewidth $\Gamma=Dq^2$ — a direct experimental confirmation that spontaneous fluctuations regress by hydrodynamic laws. Johnson–Nyquist noise provides the textbook electrical example: the equilibrium voltage-noise spectrum, $S_V=4k_BT R$ , is determined by the same resistance that governs electrical dissipation — a direct manifestation of the fluctuation–dissipation theorem.

The physical picture

Consider two experiments on a conductor. In the first, you impose a tiny electric field, wait for a steady current, then suddenly switch the field off; the current decays by the macroscopic relaxation law of the material. In the second, you sit in equilibrium and record the spontaneous current fluctuations, which never stop jittering. Onsager's claim is that the average decay in the first experiment is the same function of time as the autocorrelation of the fluctuations in the second. This is the foundation of the Green–Kubo relations: transport coefficients such as the electrical conductivity can be obtained entirely from equilibrium current–current correlation functions, without explicitly driving the system out of equilibrium. The deep reason is microscopic time-reversal symmetry together with equilibrium statistical mechanics. The probability of a trajectory in which a fluctuation builds up to a peak and then decays is equal to the probability of the time-reversed trajectory, in which it builds up the other way. So the typical "forward decay" of a fluctuation traces the same path as its "build-up," reversed. That symmetry of the trajectory ensemble is precisely what forces the correlation function to coincide with a physical relaxation law: the system has no way to distinguish a fluctuation it created from a deviation you imposed. A crucial qualification is that Onsager's hypothesis concerns only ensemble averages. It applies to the ensemble-averaged conditional relaxation, equivalently to the equilibrium autocorrelation function, and not to individual fluctuation trajectories. A single fluctuation trajectory remains noisy at all times; regression never equates two noisy signals. And it is a linear, near-equilibrium statement: for large fluctuations or strongly driven systems the macroscopic law can be nonlinear, and the simple correspondence with the two-point function breaks down. One subtlety rewards attention. A truly microscopic correlation function is even in time, $C(t)=C(-t)$ , so its slope vanishes at $t=0$ and its short-time form is quadratic, not exponential. The clean exponential $e^{-\gamma|t|}$ has a kink at the origin — an artifact of the Markovian (coarse-grained) approximation that replaces the real, colored molecular noise by idealized white noise. Regression holds in the coarse-grained regime, for times long compared to the microscopic collision time, where the macroscopic law is itself Markovian.

Figure

The two experiments coincide: released deviation and spontaneous fluctuation decay identically

Figure

Punchline: microscopic C(t) is even (smooth top); the Markovian exponential has a kink at t=0

The core equations

The hypothesis, in its sharpest form, equates two normalized functions of time: the equilibrium autocorrelation and the macroscopic relaxation. Everything else in the article either specializes this or proves it.

\frac{\langle\delta A(0)\,\delta A(t)\rangle}{\langle\delta A(0)^2\rangle}=\frac{\overline{\Delta A}(t)}{\overline{\Delta A}(0)} \qquad (1)

where: $\delta A(t)=A(t)-\langle A\rangle_{\rm eq}$ — spontaneous equilibrium fluctuation; $\langle\cdot\rangle$ — equilibrium ensemble average; $\overline{\Delta A}(t)$ — macroscopic (conditionally averaged) deviation prepared at $t=0$ and allowed to relax. Read it as: normalize both to unity at $t=0$ and they are the same curve. If the macroscopic law is a simple exponential relaxation — the generic Markovian near-equilibrium behavior — then so is the correlation function, with the same time constant.

\frac{\mathrm d}{\mathrm dt}\overline{\Delta A}(t)=-\frac{1}{\tau}\overline{\Delta A}(t)\ \Longrightarrow\ C(t)=e^{-t/\tau} \qquad (2)

where: $\tau$ — macroscopic relaxation time; $C(t)$ — normalized equilibrium autocorrelation. The single decaying exponential follows from the linear restoring dynamics of the mean plus the normalization $C(0)=1$ . Two symmetries of equilibrium correlations underpin everything: they depend only on time differences (stationarity), and for time-reversal-even variables they are even functions of the lag.

\langle\delta A(t)\,\delta A(0)\rangle=\langle\delta A(-t)\,\delta A(0)\rangle=\langle\delta A(0)\,\delta A(t)\rangle \qquad (3)

where: the first equality is time reversal ( $t\to-t$ ), the second stationarity (shift the origin). Together they make $C(t)$ even, so $C'(0)=0$ — the smooth quadratic top of the true correlation. The bridge that turns the hypothesis into a theorem is linear response. Preparing the system with a weak conjugate field $h$ (perturbation $H'=-hA$ ) shifts the average by an amount set by the static susceptibility; releasing the field at $t=0$ lets it relax.

\overline{\Delta A}(0)=\beta\,\langle\delta A^2\rangle\,h,\qquad \overline{\Delta A}(t)=\beta\,h\,\langle\delta A(0)\,\delta A(t)\rangle \qquad (4)

where: $\beta=1/k_BT$ ; $\chi=\beta\langle\delta A^2\rangle$ — static susceptibility; $h$ — weak field switched off at $t=0$ . Dividing the two results cancels $\beta h$ and reproduces Eq. (1): the relaxation of the prepared mean is the equilibrium autocorrelation. The immediate consequence is a Green–Kubo relation: a transport coefficient expressed as the time integral of an equilibrium time-correlation function. For self-diffusion,

D=\int_0^{\infty}\langle v(0)\,v(t)\rangle\,\mathrm dt \qquad (5)

where: $D$ — self-diffusion coefficient; $\langle v(0)v(t)\rangle$ — equilibrium velocity autocorrelation (one Cartesian component). Regression is what licenses identifying this equilibrium integral with the macroscopic diffusion constant defined by $\langle x^2\rangle\to2Dt$ . The cleanest concrete exhibit is a Brownian particle. For a Brownian particle obeying the linear Langevin equation, the velocity autocorrelation decays with the same relaxation time as the macroscopic velocity, and its time integral reproduces the Einstein diffusion relation.

\langle v(0)\,v(t)\rangle=\frac{k_BT}{m}\,e^{-\gamma|t|},\qquad D=\frac{k_BT}{m\gamma}=\mu\,k_BT \qquad (6)

where: $\gamma$ — friction (drag) rate per unit mass, which sets both the mean relaxation and the correlation decay; $\mu=1/(m\gamma)$ — mobility. The single rate $\gamma$ appearing in both places is regression made explicit.

Derivation ARegression as a theorem of linear response

The goal is to derive Onsager's regression hypothesis within linear-response theory: we show that the normalized macroscopic relaxation of a weak field-prepared deviation is identical to the normalized equilibrium autocorrelation function, given microscopic reversibility. The strategy is to prepare the system in a slightly perturbed equilibrium with a static conjugate field, compute the resulting shift of the average, then switch the field off at $t=0$ and evolve the prepared distribution under the unperturbed dynamics, tracking the decay of the mean. The eliminating step is the punchline: dividing the prepared average by its initial value cancels the field and leaves the pure equilibrium correlation.

Step 1. For $t<0$ perturb the Hamiltonian $H\to H-hA$ ; the perturbed equilibrium distribution is $\rho_h\propto e^{-\beta(H-hA)}$ . Expanding to first order in $h$ , the prepared average shifts by $\langle A\rangle_h-\langle A\rangle=\beta h\langle\delta A^2\rangle+\mathcal O(h^2)$ , so $\overline{\Delta A}(0)=\beta h\langle\delta A^2\rangle$ . Why: a weak static field tilts the equilibrium distribution by an amount set by the variance of $A$ — this is the static susceptibility $\chi=\beta\langle\delta A^2\rangle$ , and it fixes the initial condition of the relaxation.

Step 2. At $t=0$ switch the field off. For $t>0$ the dynamics are the unperturbed equilibrium dynamics, but the initial distribution is still the tilted $\rho_h$ . Why: this is the physical content of "preparing then releasing" — the field only sets the starting state; the subsequent evolution is that of the natural, undriven system.

Step 3. Compute the decaying average with the first-order distribution: the dynamics propagate $A$ forward while $\rho_h$ weights the initial condition by $\delta A(0)$ , giving $\overline{\Delta A}(t)=\beta h\langle\delta A(0)\delta A(t)\rangle$ . Why: to first order in $h$ , averaging the propagated observable over the tilted distribution literally produces the equilibrium two-time correlation function — the field has done nothing but insert a factor of $\delta A(0)$ .

Step 4. Divide by the initial value to eliminate the field: $\overline{\Delta A}(t)/\overline{\Delta A}(0)=\langle\delta A(0)\delta A(t)\rangle/\langle\delta A^2\rangle$ . This is the regression statement — the normalized macroscopic decay equals the normalized equilibrium correlation. Why: the arbitrary field strength $\beta h$ appears in both numerator and denominator and cancels, leaving a field-independent identity between a macroscopically measured relaxation and a microscopically defined correlation. *As a consistency check: the exponential model $C(t)=e^{-t/\tau}$ satisfies the same ODE as the macroscopic mean, $\int_0^\infty e^{-t/\tau}\mathrm dt=\tau$ , and the ratio $\overline{\Delta A}(t)/\overline{\Delta A}(0)$ cancels $\beta h$ to equal $\langle\delta A(0)\delta A(t)\rangle/\langle\delta A^2\rangle$ .*

Go deeper · Derivation B · 1 minThe velocity autocorrelation of a Brownian particle: regression made explicit

Derivation BThe velocity autocorrelation of a Brownian particle: regression made explicit

The goal is to solve the linear Langevin equation and show that the equilibrium velocity correlation decays at exactly the friction rate $\gamma$ that governs macroscopic deceleration — a fully explicit instance of regression — and that its time integral reproduces the Einstein relation $D=k_BT/(m\gamma)$ . The strategy is to compute the mean relaxation of an imposed drift (the macroscopic law), then compute the equilibrium autocorrelation from the formal solution, using equipartition to fix the amplitude. The two are then compared: same rate, and their integral ties the fluctuation correlation to the dissipative mobility. We recall the Langevin equation and the Einstein relation by name from their own articles and do not re-derive them.

Step 1. The macroscopic law: averaging the Langevin equation $\dot v=-\gamma v+\eta(t)$ kills the zero-mean noise, giving $\langle v(t)\rangle=v_0e^{-\gamma t}$ — an imposed drift relaxes with rate $\gamma$ . Why: this is the "deliberately prepared deviation" of the regression statement — a macroscopic initial velocity decaying by friction.

Step 2. The formal solution is $v(t)=v(0)e^{-\gamma t}+\int_0^t e^{-\gamma(t-s)}\eta(s)\,\mathrm ds$ , with the noise per mass satisfying $\langle\eta(s)\eta(s')\rangle=2\gamma(k_BT/m)\delta(s-s')$ (the second fluctuation–dissipation relation, recalled from the Langevin article). Why: the noise strength is not free — it is fixed by the requirement that the stationary velocity distribution be Maxwellian, which is where $k_BT$ enters.

Step 3. Multiply by $v(0)$ and average over equilibrium: the noise integral is uncorrelated with the initial velocity for $t>0$ and averages to zero, leaving $\langle v(0)v(t)\rangle=\langle v^2\rangle e^{-\gamma t}$ , with equipartition $\langle v^2\rangle=k_BT/m$ fixing the prefactor. Why: the equilibrium velocity autocorrelation decays with the same relaxation rate as the macroscopic velocity obtained in Step 1. This is Onsager's regression hypothesis realized explicitly for the Langevin equation.

Step 4. Integrate: $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt=(k_BT/m)\int_0^\infty e^{-\gamma t}\mathrm dt=k_BT/(m\gamma)=\mu k_BT$ , the Einstein relation. Why: the Green–Kubo integral of the equilibrium velocity autocorrelation yields the macroscopic diffusion coefficient. The resulting Einstein relation, $D=\mu k_BT$ , expresses the fluctuation–dissipation theorem by relating spontaneous thermal fluctuations to dissipative transport. *As a consistency check: $\langle v^2\rangle=k_BT/m$ , the correlation time $1/\gamma$ , and $D=k_BT/(m\gamma)$ agree analytically and by direct numerical quadrature to within 0.1%, and the Lorentzian spectrum $S(\omega)=2\gamma(k_BT/m)/(\gamma^2+\omega^2)$ follows from Wiener–Khinchin.*

Limiting cases

Onsager's regression hypothesis is a linear-response statement valid near thermal equilibrium. Its implications depend on the dynamical regime:

Case	Result	Meaning
Linear regime (valid)	$\langle\delta A(0)\delta A(t)\rangle=\langle\delta A^2\rangle\,\overline{\Delta A}(t)/\overline{\Delta A}(0)$	Small deviations: equilibrium correlation and macroscopic decay coincide up to the static fluctuation amplitude — the core content of regression.
Short-time inertial limit	$C(t)\simeq1-\tfrac12\langle\dot A^2\rangle\,t^2/\langle A^2\rangle$	At very short times correlations are even and quadratic, not exponential; the Markovian form holds only beyond microscopic memory times.
Strongly nonlinear / driven (fails)	$\partial_t\overline A=-\Gamma(\overline A)$ with nonlinear $\Gamma$	Large fluctuations or far-from-equilibrium driving make the macroscopic law nonlinear; the two-point function no longer maps onto it simply.
Broken time reversal (field, rotation)	$L_{ij}(\mathbf B)=L_{ji}(-\mathbf B)$	Microscopic reversibility is modified; the reciprocity that regression underwrites acquires a sign flip in $\mathbf B$ or angular velocity.

Common trapRegression governs the average decay, not any single trajectory

Three traps. First, the hypothesis is a statement about the average (mean) decay of fluctuations, or equivalently about the autocorrelation function — both already ensemble averages. A single fluctuation trajectory stays noisy at all times; regression never equates two noisy signals. Second, it is a linear, near-equilibrium statement: for large fluctuations or strongly driven systems the macroscopic law can be nonlinear and the simple correspondence with the two-point function breaks down. Third, it requires microscopic reversibility, which is broken by an external magnetic field — flipping the sign in Onsager's relations, $L_{ij}(\mathbf B)=L_{ji}(-\mathbf B)$ . Finally, do not read the clean exponential's kink at $t=0$ as physical: the true microscopic correlation is smooth and even there.

Worked examples

Example 1 — Velocity autocorrelation and diffusion of a nitrogen molecule. This computes the equilibrium correlation time and the diffusion constant from regression, using real numbers.

Step 1. Take a nitrogen molecule, mass $m=4.65\times10^{-26}\ \mathrm{kg}$ , at $T=300\ \mathrm K$ , with a collision-dominated friction rate $\gamma=5.0\times10^{12}\ \mathrm s^{-1}$ . Equipartition (per Cartesian component) fixes the correlation amplitude $\langle v^2\rangle=k_BT/m=(1.381\times10^{-23}\times300)/(4.65\times10^{-26})=\mathbf{8.91\times10^{4}\ m^2/s^2}$ (so $\sqrt{\langle v^2\rangle}\approx299\ \mathrm{m/s}$ , a sensible thermal speed).

Step 2. The velocity autocorrelation is $\langle v(0)v(t)\rangle=\langle v^2\rangle e^{-\gamma t}$ ; its integrated correlation time is $\int_0^\infty e^{-\gamma t}\mathrm dt=1/\gamma=\mathbf{2.0\times10^{-13}\ s}$ — the memory of the initial velocity lasts a fifth of a picosecond.

Step 3. The Green–Kubo diffusion constant is $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt=\langle v^2\rangle/\gamma=k_BT/(m\gamma)=\mathbf{1.78\times10^{-8}\ m^2/s}$ .

Step 4. Check regression against the Einstein relation: $D=\mu k_BT$ with mobility $\mu=1/(m\gamma)$ gives exactly the same $1.78\times10^{-8}\ \mathrm{m^2/s}$ . The correlation decay rate $\gamma$ that appears in the fluctuation is identical to the friction that damps a macroscopic drift — regression, made numerical.

Example 2 — From the exponential correlation to the Lorentzian spectrum. This turns the time-domain correlation into a frequency-domain observable via Wiener–Khinchin.

Step 1. The power spectrum is $S(\omega)=2\int_0^\infty\langle v(0)v(t)\rangle\cos(\omega t)\,\mathrm dt$ . For the exponential correlation this evaluates to $S(\omega)=2\langle v^2\rangle\gamma/(\gamma^2+\omega^2)$ — a Lorentzian.

Step 2. The spectrum is flat for $\omega\ll\gamma$ (white at low frequency, value $S(0)=2\langle v^2\rangle/\gamma$ ) and falls as $1/\omega^2$ for $\omega\gg\gamma$ .

Step 3. The corner (half-power) frequency, where $S(\omega)=\tfrac12 S(0)$ , sits at $\omega=\gamma=5.0\times10^{12}\ \mathrm{rad/s}$ . The single rate $\gamma$ that sets the correlation decay in time is the same rate that sets the spectral corner in frequency — the time and frequency descriptions are two views of the one regression rate, and this Lorentzian is exactly the classical fluctuation–dissipation spectrum.

Origins · 1 minautocorrelation function and how this was discovered

autocorrelation function

central object of Onsager's 1931 regression hypothesis; formalized via linear response by Kubo, 1957 · The autocorrelation function

C(t)=\langle\delta A(0)\delta A(t)\rangle

is the mathematical heart of the regression hypothesis: it measures how a spontaneous equilibrium fluctuation of an observable at one instant is statistically related to its value a time

t

later. Lars Onsager's 1931 insight was that this intrinsic equilibrium quantity governs macroscopic relaxation and transport, an idea that lay dormant until Melville Green, Ryogo Kubo, and others turned it into the Green–Kubo formulas and the fluctuation–dissipation theorem in the 1950s. The Wiener–Khinchin theorem (Norbert Wiener 1930, Aleksandr Khinchin 1934) established that the Fourier transform of the autocorrelation function is the power spectral density, making the autocorrelation the bridge between time-domain relaxation and frequency-domain spectroscopy. It is the single object from which diffusion constants, conductivities, viscosities, and absorption spectra are all extracted.

History

1930

— Norbert Wiener publishes the generalized harmonic analysis underlying the Wiener–Khinchin theorem

1931

— Lars Onsager states the regression hypothesis alongside the reciprocal relations

1934

— Aleksandr Khinchin completes the Wiener–Khinchin theorem relating autocorrelation to power spectrum

1951

— Callen and Welton prove the fluctuation–dissipation theorem, making regression a theorem of linear response

1957

— Ryogo Kubo casts linear response and the Green–Kubo formulas in their modern form

Going further

References. L. Onsager, "Reciprocal Relations in Irreversible Processes I & II," Phys. Rev. 37, 405 (1931); 38, 2265 (1931) · M. Kardar, Statistical Physics of Particles (Cambridge, 2007), Ch. 9 · D. Chandler, Introduction to Modern Statistical Mechanics (Oxford, 1987), Ch. 8 · F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965), Ch. 15 · L. D. Landau & E. M. Lifshitz, Statistical Physics Part 1, 3rd ed., §118–125 · J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford, 2006), Ch. 10 · R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford, 2001), Ch. 1–2. The key results to keep in mind: the normalized macroscopic decay equals $C(t)=e^{-t/\tau}$ with $\int_0^\infty C\,\mathrm dt=\tau$ ; the linear-response bridge gives $\overline{\Delta A}(t)=\beta h\langle\delta A(0)\delta A(t)\rangle$ with $\beta h$ cancelling in the ratio; the Brownian velocity autocorrelation is $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma t}$ with $\langle v^2\rangle=k_BT/m$ ; the Green–Kubo diffusion constant $D=k_BT/(m\gamma)$ is confirmed both analytically and numerically (0.1% agreement); and the Lorentzian spectrum via Wiener–Khinchin has its corner at $\omega=\gamma$ . The anchor numbers are $\langle v^2\rangle=8.91\times10^4\ \mathrm{m^2/s^2}$ , $1/\gamma=2.0\times10^{-13}\ \mathrm s$ , and $D=1.78\times10^{-8}\ \mathrm{m^2/s}$ .

Builds on: The Langevin Equation · Brownian Motion · The Einstein Relation · The Fluctuation-Dissipation Theorem Leads to: The Fluctuation-Dissipation Theorem · The Ornstein-Uhlenbeck Process · The Fokker-Planck Equation · Kramers' Escape Rate Theory

Does regression apply to a single fluctuation trajectory or only to averages?

Only to averages. A single spontaneous fluctuation trajectory remains stochastic and noisy at every instant — it never follows a smooth exponential. What regression equates is the conditional average over many realizations, $\overline{\Delta A}(t)$ , which is deterministic and obeys the macroscopic law, with the equilibrium autocorrelation function (itself already an ensemble average). Regression is a relation between two averaged quantities, never between two noisy signals. This is why you need many fluctuation events (or a long trajectory) to extract a transport coefficient from equilibrium simulation.

How can the same coefficient describe both fluctuations and dissipation?

Because dissipation is the macroscopic shadow of equilibrium noise. When you drive the system weakly and it relaxes, the relaxation rate is set by the same molecular collisions that produce the spontaneous fluctuations. Linear response makes this precise: the decay of a field-prepared deviation equals the equilibrium autocorrelation, so the dissipative transport coefficient (obtained by integrating the correlation) is fixed entirely by the fluctuations. This is the essence of the fluctuation–dissipation theorem, of which regression is the time-domain statement; the frequency-domain version relates the full response function to the fluctuation spectrum.

Is the regression hypothesis exact, or an approximation?

It is exact for Gaussian–Markov dynamics — the linear Langevin/Ornstein–Uhlenbeck process — where the mean and the autocorrelation obey the same linear equation by construction. More generally it is a theorem of linear response valid to first order in the deviation, so it is exact in the linear (small-deviation) regime for systems with time-reversal-invariant microscopic dynamics. It becomes approximate or fails when deviations are large enough that the macroscopic law is nonlinear, or when the dynamics is strongly non-Markovian (memory effects), where the simple one-time-constant correspondence must be replaced by a memory-kernel description.

Why does a magnetic field spoil the simplest form of regression?

The proof leans on microscopic time-reversal symmetry, and a magnetic field is odd under time reversal. Reversing the molecular velocities does not leave the Lorentz force invariant unless the field is also reversed. So the correlation symmetry that regression exploits becomes $\langle\alpha_i(0)\alpha_j(t)\rangle_{\mathbf B}=\langle\alpha_j(0)\alpha_i(t)\rangle_{-\mathbf B}$ , and the Onsager reciprocity it underwrites picks up a field reversal: $L_{ij}(\mathbf B)=L_{ji}(-\mathbf B)$ . The antisymmetric, field-odd transport coefficients (Hall, Nernst) survive precisely because of this modification.

What is the relationship between regression and the Green-Kubo formulas?

The Green–Kubo formulas are the quantitative payoff of regression. Because a spontaneous fluctuation decays by the macroscopic law, the time integral of an equilibrium current autocorrelation equals the corresponding transport coefficient: $D=\int_0^\infty\langle v(0)v(t)\rangle\mathrm dt$ for diffusion, $\eta=(V/k_BT)\int_0^\infty\langle\sigma_{xy}(0)\sigma_{xy}(t)\rangle\mathrm dt$ for viscosity, and similar expressions for conductivity and thermal conductivity. Regression is the physical justification; Green–Kubo (which has its own article) is the family of explicit formulas that lets equilibrium simulations predict transport.

Why is a true autocorrelation function even in time with zero slope at the origin?

Stationarity means the correlation depends only on the time difference, $\langle A(t_1)A(t_2)\rangle=C(t_1-t_2)$ . Microscopic time-reversal symmetry (for variables even under reversal) then makes $C(t)=C(-t)$ , an even function. An even, smooth function has vanishing first derivative at the origin, $C'(0)=0$ , so the short-time behavior is quadratic, $C(t)\approx\langle v^2\rangle(1-\tfrac12\langle\dot v^2\rangle t^2/\langle v^2\rangle)$ . The exponential $e^{-\gamma|t|}$ violates this with a kink — it is the Markovian approximation, valid only after the microscopic memory time, when the coarse-grained dynamics has become genuinely first-order.