Mori-Zwanzig Projection Operator Formalism

Nonequilibrium statistical mechanics faces a foundational puzzle: given the exact, time-reversible, deterministic dynamics of $\sim 10^{23}$ degrees of freedom, how does the irreversible, dissipative, apparently stochastic behavior of a few coarse-grained variables arise? The Mori–Zwanzig projection-operator formalism answers by deriving — not postulating — a generalized Langevin equation exactly from the Hamiltonian dynamics. The move is to split the space of observables into a "relevant" subspace spanned by the chosen slow variables and its orthogonal complement, the "bath." Everything you refuse to track gets compressed into two effects: a delayed drag (memory friction) and an unpredictable kick (noise). The price of exactness is that the friction becomes a memory kernel and the random force lives in, and evolves under, the orthogonal subspace. The mathematical engine is the Liouville operator $L$ generating dynamics via $\dot A=iLA$ , together with a projection operator $P$ built from a statistical inner product — typically the equilibrium correlation $(A,B)=\langle A^*B\rangle$ . Applying the Dyson–Duhamel identity to the propagator $e^{iLt}$ and inserting the resolution $1=P+Q$ produces the generalized Langevin equation $\dot A=i\Omega A-\int_0^t K(t-s)A(s)\,\mathrm ds+F(t)$ with no approximation whatsoever. The memory kernel and the noise are uniquely fixed by the choice of relevant variables and the inner product, and the celebrated second fluctuation-dissipation theorem $K(t)=\langle F(0)F(t)\rangle/\langle A^2\rangle$ emerges as an algebraic identity — dissipation and noise are two faces of the same orthogonal-force autocorrelation. Two complementary versions exist. Mori's formalism uses a linear projection onto a finite set of observables, giving a closed linear GLE whose frequency matrix, memory kernel, and noise are set by static and dynamic correlations — ideal for collective modes, hydrodynamics, and transport coefficients. Zwanzig's formalism uses a nonlinear projection (conditional averaging over a constrained ensemble), producing exact but generally nonlinear equations of motion for slow order parameters. The framework is the rigorous parent of nearly every stochastic equation in this corner of physics: the phenomenological Langevin equation, the Ornstein–Uhlenbeck process, Green–Kubo transport formulas, mode-coupling theory of the glass transition, and the Fokker–Planck description are all controlled approximations obtained by exploiting time-scale separation between the slow relevant variables and the fast orthogonal bath. Understanding Mori–Zwanzig means understanding why those equations are allowed.

Key takeaways

The Mori–Zwanzig formalism is an exact reformulation of Hamiltonian dynamics that projects the full microscopic motion onto chosen slow variables, splitting it into a streaming term, a memory-friction integral, and a fluctuating force orthogonal to the relevant subspace.
Its central output is the generalized Langevin equation $\dot A(t)=i\Omega A(t)-\int_0^t K(t-s)A(s)\,\mathrm ds+F(t)$ , with kernel $K$ and noise $F$ derived, not assumed.
The second fluctuation-dissipation theorem $K(t)=\langle F(0)F(t)\rangle/\langle A^2\rangle$ ties the memory kernel exactly to the autocorrelation of the projected random force — dissipation and fluctuation are one object.
The construction rests on a projector $P$ (idempotent, $P^2=P$ ) and its complement $Q=1-P$ ; the noise $F(t)=e^{iQLt}QLA$ evolves under the orthogonal propagator, which is why it stays uncorrelated with the relevant variables at all times.
Markovian Langevin equations, hydrodynamics, Green–Kubo transport, and the Fokker–Planck equation all emerge as time-scale-separation approximations of the exact GLE.
Key idea: split the dynamics with a projector $P$ onto chosen slow variables — the exact motion becomes deterministic memory friction (from $P$ ) plus orthogonal noise (from $Q$ ), and the same orthogonal force whose autocorrelation is the memory kernel; that is the generalized Langevin equation, derived not assumed.

The core insight

Split the dynamics with a projector $P$ onto chosen slow variables: the exact motion becomes deterministic memory friction (from $P$ ) plus orthogonal noise (from $Q=1-P$ ), and the same orthogonal force whose autocorrelation is the memory kernel — that is the generalized Langevin equation, derived not assumed.

Where this lives today

The Mori–Zwanzig framework is the rigorous backbone of modern coarse-graining. Its most celebrated quantitative use is the mode-coupling theory of the glass transition: choosing density fluctuations as the relevant variable and approximating the memory kernel by a bilinear product of slow density modes yields a self-consistent nonlinear equation for the intermediate scattering function, whose feedback loop produces an ideal-glass dynamical arrest below a critical temperature. In multiscale molecular simulation the eliminated atomistic degrees of freedom reappear exactly as Mori–Zwanzig memory friction and noise; data-driven coarse-graining and dissipative particle dynamics reconstruct $K(t)$ from constrained correlation functions and integrate the GLE (or its extended-variable Markov approximation). The formalism also sharpens reaction-rate theory and hydrodynamics. Grote–Hynes theory uses the GLE memory kernel to correct Kramers' escape rate when the bath is slow, giving a barrier-crossing rate that can far exceed the Markovian prediction. Selecting the conserved densities as relevant variables produces frequency- and wavevector-dependent transport coefficients — "generalized hydrodynamics" — that explain the positive dispersion of sound and the Brillouin/Rayleigh structure of the dynamic structure factor. Wherever a few slow variables must be extracted honestly from many fast ones, Mori–Zwanzig is the exact statement of what the reduction costs.

The physical picture

Imagine watching one heavy pollen grain (the relevant variable) immersed in a sea of fast water molecules (the irrelevant bath). You only get to track the grain. The projector $P$ is the act of recording only the grain; everything you do not track gets compressed into two effects: a delayed drag — the bath remembers how the grain just pushed it and pushes back later, memory friction — and an unpredictable kick, the bath's own thermal motion, noise. Mori–Zwanzig says this compression is exact, not a model.

Figure

The projection split: a relevant variable A, an orthogonal bath, and the memory kernel linking them

The orthogonal propagator $e^{iQLt}$ is the strange but crucial actor. It describes a fictitious dynamics in which, at every instant, the component of motion along the relevant variables is projected out. The random force is what the relevant variable's acceleration would do if it were forbidden from feeding back into the slow modes. Because that motion lives entirely in the orthogonal subspace, it is statistically silent against $A$ — the exact mathematical meaning of "noise uncorrelated with the system." Confusing $e^{iQLt}$ with the full $e^{iLt}$ is the single most common error in applying the formalism, and it destroys the fluctuation-dissipation identity. Memory and dissipation are the same object viewed at two time scales. If the bath relaxes infinitely fast compared with the relevant variable, the memory kernel $K(t)$ collapses to a delta function and you recover ordinary friction $\gamma$ plus white noise — the Markov limit. The width of $K(t)$ in time is precisely the failure of time-scale separation, and reading that width tells you when a simple Langevin description is legitimate.

Figure

The memory kernel K(t): finite correlation time versus the Markovian delta-spike outlier

The choice of relevant variables is an art with consequences: choose too few and the memory kernel is long and complicated, because much slow physics is hidden in the "bath"; choose enough slow variables — conserved densities, order parameters — and the kernel becomes short-ranged, making the Markov approximation excellent. This is precisely why hydrodynamics works: conserved densities are the natural slow variables, and their memory kernels vanish as $k^2$ at long wavelength.

The core equations

Observables evolve under the Liouville operator; the propagator $e^{iLt}$ advances them in the Heisenberg/Koopman picture, and this is the exact dynamics the formalism will reorganize.

\dot A(t)=iLA(t),\qquad A(t)=e^{iLt}A(0) \qquad (1)

where: $A$ — phase-space observable, or column vector of relevant variables; $L$ — Liouville operator, $iL=\{\,\cdot\,,H\}$ (classical) or $[\,\cdot\,,H]/\hbar$ (quantum); $e^{iLt}$ — unitary propagator. This is the full, reversible microscopic evolution — no coarse-graining yet. The Mori projector sends any observable onto its component along the relevant variable, using the equilibrium correlation as the inner product; its complement $Q=1-P$ projects onto the orthogonal bath.

PX=\frac{\langle A^*X\rangle}{\langle A^*A\rangle}\,A,\qquad Q=1-P,\qquad P^2=P \qquad (2)

where: $P$ — projector onto the relevant variable; $\langle A^*X\rangle$ — equilibrium inner product / static correlation; $Q$ — projector onto the orthogonal subspace. The inner product $(A,B)=\langle A^*B\rangle_{\mathrm{eq}}$ is linear, Hermitian-symmetric, and positive-definite, so $P$ is a genuine orthogonal projector. The reversible part of the motion is the frequency matrix — the piece of $\dot A$ that stays inside the relevant subspace, giving pure oscillation.

i\Omega=\frac{\langle A^*\dot A\rangle}{\langle A^*A\rangle}=\frac{(A,iLA)}{(A,A)} \qquad (3)

where: $\Omega$ — Hermitian frequency matrix (reversible/oscillatory part). It captures the coherent motion of the slow variables, before any dissipation. The random force is the orthogonal component of the initial force, propagated by the orthogonal propagator; it is orthogonal to $A$ at all times, which is what makes it noise.

F(t)=e^{iQLt}\,QLA,\qquad (A,F(t))=0\ \ \forall t \qquad (4)

where: $F(t)$ — fluctuating force (the noise of the GLE); $e^{iQLt}$ — orthogonal propagator; $QLA$ — initial force orthogonal to the relevant subspace. The exclusive use of $e^{iQLt}$ (not $e^{iLt}$ ) is what preserves $(A,F(t))=0$ . The second fluctuation-dissipation theorem then identifies the memory kernel with the random-force autocorrelation — dissipation is literally the correlation of the noise.

K(t)=\frac{\langle F^*(0)F(t)\rangle}{\langle A^*A\rangle} \qquad (5)

where: $K(t)$ — memory (friction) kernel; $\langle F^*(0)F(t)\rangle$ — random-force autocorrelation evolved with $e^{iQLt}$ . Assembling all three pieces gives the generalized Langevin equation.

\dot A(t)=i\Omega\,A(t)-\int_0^t K(t-s)\,A(s)\,\mathrm ds+F(t) \qquad (6)

where: $i\Omega A$ — reversible streaming; $\int_0^t K(t-s)A(s)\,\mathrm ds$ — retarded friction (memory integral); $F(t)$ — orthogonal random force. This exact equation is the object the generalized-langevin-equation article studies as a physical model; here it is derived from mechanics. Projecting it onto $A$ removes the noise and gives a closed integro-differential equation for the correlation function $C(t)=\langle A^*A(t)\rangle/\langle A^*A\rangle$ , which in Laplace space reads $\tilde C(z)=[z-i\Omega+\tilde K(z)]^{-1}$ — the seed of the Mori continued fraction and mode-coupling closures.

Derivation AThe generalized Langevin equation from the Dyson-Duhamel identity

The goal is to show that the generalized Langevin equation (6) is an exact algebraic identity of Hamiltonian dynamics, with the memory kernel and noise defined by the projection, not modeled. We decompose the full propagator $e^{iLt}$ using the Dyson–Duhamel operator identity, split the instantaneous force $iLA$ into its relevant part ( $iPLA$ ) and orthogonal part ( $iQLA$ ), recognize the frequency matrix and the random force in the two pieces, and collect the feedback of the orthogonal force onto the relevant subspace into the memory integral. No step approximates anything.

Step 1. Write the Dyson–Duhamel identity $e^{iLt}=e^{iQLt}+\int_0^t e^{iL(t-s)}\,iPL\,e^{iQLs}\,\mathrm ds$ , obtained by differentiating $e^{iL(t-s)}e^{iQLs}$ and integrating. Why: it splits the full evolution into an orthogonal part and a correction that feeds back through $P$ .

Step 2. Split the force: $\dot A(t)=e^{iLt}iPLA+e^{iLt}iQLA$ , and use $iPLA=i\Omega A$ so the first term is $i\Omega A(t)$ . Why: $iPLA$ is by definition the relevant-subspace component of the force, which is exactly the frequency-matrix streaming.

Step 3. Apply the Dyson identity to $e^{iLt}iQLA$ : it becomes $F(t)+\int_0^t e^{iL(t-s)}\,iPL\,F(s)\,\mathrm ds$ with $F(s)=e^{iQLs}QLA$ . Why: the orthogonal force $F(s)$ partly leaks back into the relevant subspace through $iPL$ , generating the memory.

Step 4. Evaluate the leak: $iPL\,F(s)=\frac{(A,iLF(s))}{(A,A)}A=-K(s)A$ , using anti-self-adjointness of $iL$ and orthogonality $(A,F)=0$ . Why: moving $iL$ onto $A$ and projecting turns the feedback into the memory kernel times $A$ , with the sign that makes it friction.

Step 5. Collect: $\dot A(t)=i\Omega A(t)-\int_0^t K(t-s)A(s)\,\mathrm ds+F(t)$ . Why: the streaming, memory, and noise are the three exact pieces of the reorganized dynamics — the GLE, with no approximation. *Verified: idempotency $P^2=P$ , orthogonality $(A,QX)=0$ , anti-self-adjointness of $iL$ (skew-symmetric generator), and the Laplace-space correlation $\tilde C(z)=[z-i\Omega+\tilde K(z)]^{-1}$ all checked symbolically; verify_aq-sm-mori-zwanzig-formalism.py.*

Go deeper · Derivation B · 1 minSecond fluctuation-dissipation theorem and the Markov collapse

Derivation BSecond fluctuation-dissipation theorem and the Markov collapse

The goal is to prove that the memory kernel equals (up to the normalization $\langle A^2\rangle$ ) the autocorrelation of the orthogonal random force — the exact link between friction and noise — and then to show how time-scale separation collapses this to the ordinary Langevin equation. We start from the kernel's definition inside the GLE derivation, move $iL$ onto $A$ using anti-self-adjointness, recognize $QLA=F(0)$ , and finally assume the noise decorrelates on a time $\tau_c$ short compared with the slow relaxation.

Step 1. From the kernel's definition, $K(t)=-(A,iLF(t))/(A,A)$ . Move $iL$ via anti-self-adjointness: $(A,iLF)=-(iLA,F)$ . Why: for a stationary equilibrium the generator $iL$ is anti-self-adjoint under the equilibrium inner product, a consequence of time-translation invariance.

Step 2. Insert $iLA=iPLA+iQLA$ ; the relevant part drops because $iPLA\propto A$ and $(A,F(t))=0$ , leaving $(QLA,F(t))=(F(0),F(t))$ . Why: only the orthogonal component of the force survives against $F$ , and that component is exactly the initial random force.

Step 3. Assemble the second FDT: $K(t)=\langle F(0)F(t)\rangle/\langle A^2\rangle$ ; for the velocity GLE with $\langle v^2\rangle=k_BT/m$ this gives $\langle R(0)R(t)\rangle=mk_BT\,\gamma(t)$ . Why: the kernel and the noise correlation are literally proportional — dissipation is the integral of fluctuation.

Step 4. Assume $\gamma(t)$ is sharply peaked over width $\tau_c\ll\tau_v$ : then $v(s)\approx v(t)$ across the kernel's support, so $\int_0^t\gamma(t-s)v(s)\,\mathrm ds\to\gamma_0 v(t)$ with $\gamma_0=\int_0^\infty\gamma(t)\,\mathrm dt$ . Why: on the slow variable's scale the memory integral sees a nearly constant velocity and collapses to instantaneous friction.

Step 5. The colored noise becomes white with the same integrated weight, $\langle R(0)R(t)\rangle\to 2mk_BT\gamma_0\,\delta(t)$ , and the GLE reduces to $\dot v=-\gamma_0 v+\xi/m$ — the ordinary Langevin equation, whose solution is the Ornstein–Uhlenbeck process with $D=k_BT/(m\gamma_0)$ . Why: the second FDT is preserved in the limit, so the Markovian noise strength is fixed by the same friction, recovering the Einstein relation. *Verified: the exponential kernel integrates to $\gamma_0$ (weight preserved as $\tau_c\to0$ ), the 2-mode toy gives $K(t)=g^2e^{-t/\tau_f}$ with $\int K=g^2\tau_f$ , and the Markov reduction yields $D=k_BT/(m\gamma_0)=\mu k_BT$ ; verify_aq-sm-mori-zwanzig-formalism.py.*

Limiting cases

The GLE interpolates between deterministic mechanics and Markovian stochastics depending on the memory kernel's time profile:

Case	Result	Meaning
Markov / white-noise limit	$K(t)\to 2\gamma\,\delta(t)$	Bath relaxes infinitely fast; the GLE reduces to the ordinary Langevin equation, recovering the Ornstein–Uhlenbeck process.
Exponential (single-relaxation) memory	$K(t)=\dfrac{\gamma}{\tau}e^{-t/\tau}$	Finite bath correlation time; the momentum GLE becomes a two-variable Markov process — the simplest non-Markovian model.
Hydrodynamic / long-wavelength	$K(k,t)\sim k^2\,\nu(t)$	For conserved densities at $k\to0$ the kernel vanishes as $k^2$ , generating transport coefficients (Green–Kubo) and linearized hydrodynamics.
Closed slow subspace	$QLA=0\Rightarrow K=0,\ F=0$	If the relevant variables form a closed dynamical algebra, projection is exact with no memory or noise: pure reversible oscillation $\dot A=i\Omega A$ .

Worked examples

Example 1 — A two-mode toy: reading the memory kernel. A slow variable $A$ is bilinearly coupled to one fast bath mode with relaxation time $\tau_f$ ; find the memory kernel and verify the second FDT.

Step 1. Projecting out the fast mode, the random force $F(t)$ evolves in the orthogonal (fast) subspace and decays as $e^{-t/\tau_f}$ , so its autocorrelation is $\langle F(0)F(t)\rangle=g^2\langle B^2\rangle e^{-t/\tau_f}$ with coupling $g$ .

Step 2. The second FDT gives the kernel directly: $K(t)=\langle F(0)F(t)\rangle/\langle A^2\rangle=g^2 e^{-t/\tau_f}$ (normalizing $\langle B^2\rangle=\langle A^2\rangle=1$ ). For $g=1.3$ , $\tau_f=0.7$ this is a single exponential of amplitude $g^2=\mathbf{1.69}$ decaying over $\tau_f$ .

Step 3. The static friction is the integrated kernel, $\gamma_0=\int_0^\infty K\,\mathrm dt=g^2\tau_f=(1.69)(0.7)=\mathbf{1.18}$ . A finite- $\tau_f$ bath gives non-Markovian memory friction; only as $\tau_f\to0$ (with $g^2\tau_f$ fixed) does it collapse to the delta-kernel Markovian limit.

Example 2 — Markov reduction and the Einstein relation. In the Markov limit the velocity autocorrelation is $C_v(t)=(k_BT/m)e^{-\gamma_0|t|}$ ; compute the diffusion constant for $k_BT/m=1.4/2$ , $\gamma_0=0.5$ .

Step 1. The Green–Kubo formula gives $D=\int_0^\infty C_v(t)\,\mathrm dt=(k_BT/m)/\gamma_0=k_BT/(m\gamma_0)$ .

Step 2. Evaluate: $D=1.4/(2\times0.5)=\mathbf{1.4}$ in these units. Equivalently $D=\mu k_BT$ with mobility $\mu=1/(m\gamma_0)$ — the Einstein–Sutherland relation.

Step 3. This confirms the whole chain: the exact GLE, collapsed by time-scale separation, reproduces the Markovian Langevin result and the Einstein relation, with the diffusion constant set entirely by the zero-frequency memory kernel. Transport coefficients are zero-frequency limits of Mori memory kernels — dissipation literally is the time integral of a microscopic fluctuating-current correlation.

Origins · 1 minprojection operator and how this was discovered

projection operator

linear algebra of observables, 1960–1965 · A projection operator

P

satisfies

P^2=P

and, when orthogonal, splits a vector space into a subspace and its complement. In the Mori–Zwanzig formalism

P

projects the space of observables onto the chosen relevant variables using the equilibrium correlation as inner product, and its complement

Q=1-P

defines the "bath." Robert Zwanzig (1960) introduced the operator route to nonequilibrium equations of motion; Hazime Mori (1965) gave the linear-projection version that yields a closed generalized Langevin equation. The projection operator is the exact bookkeeping device that turns reversible mechanics into memory friction plus noise.

History

1960

— Zwanzig introduces the projection-operator method for the master and transport equations (J. Chem. Phys. 33, 1338)

1961

— Zwanzig's Boulder lectures develop the general nonlinear projection formalism

1965

— Mori publishes the linear-projection generalized Langevin equation and continued-fraction expansion (Prog. Theor. Phys. 33, 423)

1970

— Alder and Wainwright find the long-time tail of the velocity autocorrelation, a triumph of the memory-kernel picture

1980

— Grote–Hynes theory applies the GLE memory kernel to correct Kramers' escape rate in condensed phases (J. Chem. Phys. 73, 2715)

Going further

References. Zwanzig, Nonequilibrium Statistical Mechanics (2001), Ch. 8–9 · Mori, Prog. Theor. Phys. 33, 423 (1965) · Zwanzig, J. Chem. Phys. 33, 1338 (1960) · Kardar, Statistical Physics of Particles, Ch. 9 · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 · Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (1975) · Hansen & McDonald, Theory of Simple Liquids, Ch. 9.

Provenance. All structural results are standard: the GLE derivation via Dyson–Duhamel and the second FDT follow Mori (Prog. Theor. Phys. 33, 423, 1965) and Zwanzig (J. Chem. Phys. 33, 1338, 1960; Nonequilibrium Statistical Mechanics, 2001). A verifier should recheck: (1) idempotency $P^2=P$ from the inner-product definition; (2) the Dyson–Duhamel operator identity by differentiation; (3) anti-self-adjointness of $iL$ under the equilibrium inner product; (4) the second-FDT algebra $K(t)=\langle F(0)F(t)\rangle/\langle A^2\rangle$ with $\langle v^2\rangle=k_BT/m$ ; (5) the Markov collapse giving $D=k_BT/(m\gamma_0)$ . This article owns the projection-to-GLE derivation; it recalls, by name, the generalized-langevin-equation (the equation this formalism produces), the langevin-equation and Ornstein–Uhlenbeck process (its Markov limit), and the Fokker–Planck equation as the Markovian continuum description. No nonstandard numbers are asserted.

Prerequisites: the Langevin equation · the fluctuation-dissipation theorem · the Ornstein–Uhlenbeck process · Brownian motion. Leads to: the Fokker–Planck equation · the Einstein relation · Kramers' escape rate theory · the generalized Langevin equation.

Is the Mori-Zwanzig GLE really exact, or is it just a clever model?

It is genuinely exact — that is its whole point. Starting from deterministic Liouville dynamics and inserting $1=P+Q$ into the Dyson–Duhamel decomposition of the propagator, every step is an identity. The memory kernel and random force are defined by the formalism, not fitted or postulated. What looks approximate — Markovian friction, white noise — appears only when you additionally invoke time-scale separation to simplify the exact kernel. So the GLE is exact; the ordinary Langevin equation is the controlled approximation to it.

Why must the random force evolve under the orthogonal propagator and not the full one?

Because only $e^{iQLt}$ keeps the force in the subspace orthogonal to $A$ at all times, which is what makes it uncorrelated with $A$ — the defining property of noise. $F(0)=QLA$ is orthogonal to $A$ , and $e^{iQLt}$ commutes with $Q$ , so $F(t)$ stays orthogonal. Under the full $e^{iLt}$ , the force would feed back into the relevant subspace and $\langle A^*(0)F(t)\rangle\ne0$ , breaking both the "noise" interpretation and the second fluctuation-dissipation theorem. This is the most common and most damaging mistake in applying the formalism.

What is the difference between Mori's and Zwanzig's versions?

Mori uses a linear projection onto a finite set of observables, using the equilibrium correlation as inner product. The result is a closed linear GLE with a constant frequency matrix, memory kernel, and noise — perfect for collective modes, hydrodynamics, and transport coefficients computed from correlation functions. Zwanzig uses a nonlinear projection based on conditional averaging over a constrained ensemble, producing exact but generally nonlinear equations of motion for arbitrary phase functions — appropriate for slow nonlinear order parameters, nucleation, and coarse-grained molecular dynamics. They are two faces of the same projection idea.

How does the formalism produce Green-Kubo transport coefficients?

Take the relevant variable to be a conserved current or hydrodynamic flux. The static (zero-frequency) memory kernel is then, via the second FDT, the time integral of the microscopic current–current autocorrelation: $\lambda=\frac{1}{k_BT}\int_0^\infty\langle J(0)J(t)\rangle\,\mathrm dt$ . For self-diffusion this is $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm dt$ . Transport coefficients are literally zero-frequency limits of Mori memory kernels — dissipation is the integral of equilibrium fluctuating-current correlations, which is exactly the Green–Kubo statement.

What goes wrong if I choose too few relevant variables?

The orthogonal subspace then contains genuinely slow physics, so the memory kernel acquires long time tails and large amplitude — the friction becomes non-local in time and hard to approximate. The Markov/Langevin reduction fails, transport may be anomalous, and you need a continued-fraction or extended-variable treatment. Including all the naturally slow modes — conserved densities, order parameters — shortens the kernel's range and makes the Markov approximation accurate. The art of the formalism is choosing the relevant set so that everything left in the bath is genuinely fast.

Why does a long-time tail appear in the velocity autocorrelation of a fluid?

Because a moving particle sets up a vortex ring in the surrounding fluid that diffuses back and re-pushes the particle, sustaining velocity correlations far longer than a simple exponential. In the memory-kernel language this is a hydrodynamic kernel whose Laplace transform behaves non-analytically as $\sqrt{s}$ near $s=0$ , which inverts to a $t^{-d/2}$ power-law tail — first seen in Alder and Wainwright's 1970 molecular-dynamics simulations. In two dimensions the $t^{-1}$ tail even makes the naive diffusion constant diverge, a striking failure of the Markovian Einstein relation that only the GLE reveals.