Asymmetric Thermal Relaxation in Driven Systems: Rotations go Opposite Ways

It was predicted and recently experimentally confirmed that systems with microscopically reversible dynamics in locally quadratic potentials warm up faster than they cool down. This thermal relaxation asymmetry challenged the local-equilibrium paradigm valid near equilibrium. Because the intuition and proof hinged on the dynamics obeying detailed balance, it was not clear whether the asymmetry persists in systems with irreversible dynamics. To fill this gap, we here prove the relaxation asymmetry for systems driven out of equilibrium by a general linear drift. The asymmetry persists due to a non-trivial isomorphism between driven and reversible processes. Moreover, rotational motions emerge that, strikingly, occur in opposite directions during heating and cooling. This highlights that noisy systems do not relax by passing through local equilibria.

It was predicted and recently experimentally confirmed that systems with microscopically reversible dynamics in locally quadratic potentials warm up faster than they cool down.This thermal relaxation asymmetry challenged the local-equilibrium paradigm valid near equilibrium.Because the intuition and proof hinged on the dynamics obeying detailed balance, it was not clear whether the asymmetry persists in systems with irreversible dynamics.To fill this gap, we here prove the relaxation asymmetry for systems driven out of equilibrium by a general linear drift.The asymmetry persists due to a non-trivial isomorphism between driven and reversible processes.Moreover, rotational motions emerge that, strikingly, occur in opposite directions during heating and cooling.This highlights that noisy systems do not relax by passing through local equilibria.
According to the laws of thermodynamics, systems in contact with a thermal environment evolve to the temperature of their surroundings in the process called thermal relaxation [1].Relaxation close to equilibrium may be explained by linear response theory conceptually based on Onsager's regression hypothesis [2][3][4].That is, relaxation from a temperature quench is indistinguishable from the decay of a spontaneous thermal fluctuation at equilibrium [2][3][4].Analogous results were meanwhile formulated also for relaxation near non-equilibrium steady states [5][6][7].Beyond the linear regime, however, the regression hypothesis and perturbative arguments fail.
A particularly striking feature of relaxation was unraveled with the discovery of the asymmetry between heating and cooling from thermodynamically equidistant temperature quenches [57].That is, it was found that systems with locally quadratic energy landscapes and microscopically reversible dynamics heat up faster than they cool down.Later works expanded on this result [58][59][60].The asymmetry was recently quantitatively confirmed by experiments [56].
The asymmetry emerges because the entropy production within the system during heating is more efficient than heat dissipation into the environment during cooling [57].In turn, close to equilibrium they become equivalent and symmetry is restored [56,57].An even deeper understanding of the asymmetry was recently achieved by means of "thermal kinematics" [56].However, both the reasoning and the proof of the asymmetry [56,57,61] seem to hinge on the reversibility of the dynamics.Therefore, the persistence of the asymmetry in systems driven into non-equilibrium steady states (NESS) was unexpected.In particular, a non-conservative force profoundly changes relaxation behavior [62][63][64][65] even near stable fixed points [66] and in systems with linear drift [67], and may thus a priori also break the asymmetry.5) during heating from Tc (red) and cooling from T h (blue) with (solid lines) and without (dashed lines) irreversible shear flow.The shear changes D i t , but the thermal relaxation asymmetry D c t < D h t for t > 0 remains valid.Inset: Temperatures Ti before the quench are chosen thermodynamically equidistant, i.e.
Here, we investigate the speed and asymmetry of thermal relaxation to an NESS.As a paradigmatic example we first consider a harmonically confined Rouse polymer with hydrodynamic interactions and internal friction driven by shear flow (see Fig. 1), and demonstrate that heating is faster than cooling.Next we provide a systematic analysis of relaxation under broken detailed balance and explain under which conditions heating and cooling both become faster.Finally, we prove that all ergodic systems with a linear drift, including those driven arbitrarily far from equilibrium and displaying rotational motions, arXiv:2304.06702v2[cond-mat.stat-mech]12 Jun 2023 heat up faster than they cool down.In this regime the notion of a local effective non-equilibrium temperature is nominally impossible.Our proof, which exploits dualreversal symmetry, unravels a non-trivial isomorphism between reversible and driven systems.Finally, we find a new unexpected facet of the relaxation asymmetryrotational motions occur in opposite directions during heating and cooling, respectively.
Setup and motivating example.-Therelaxation asymmetry was originally proven for reversible diffusions in locally quadratic energy landscapes as well as their lowdimensional projections [57,61].It states that such systems, when quenched from thermodynamically equidistant (TED) temperatures T h , T c to an ambient temperature T w with T c < T w < T h , heat up faster than they cool down.In quantitative terms, the generalized excess free energy in units of k B T w [66,[69][70][71] or non-adiabatic entropy production [72,73] (i.e. the relative entropy in units of k B [74] between the instantaneous P w i (x, t) and stationary p w s (x) probability density at T w with i = h, c) is always smaller during heating [57,61].That is, D c t < D h t for all t > 0 and all TED T h and T c .In a strict sense, the asymmetry is to be understood as a statement about linearized drift around a local minimum in some high-dimensional energy landscape [57]; counterexamples for diffusion in rugged landscapes [57] and for small quenches also in sufficiently anharmonic wells [60] are known.The generalization to driven systems therefore involves a linear drift that, however, does not derive from a potential and breaks detailed balance.Our main result is the discovery and proof (see last section) of the asymmetry D c t < D h t in driven systems.Consider a d-dimensional system evolving according to the overdamped Langevin equation [75,76] with square drift and noise-amplitude matrices, A and σ i , respectively.In terms of the friction matrix γ, given by Stokes' law, the positive definite diffusion matrix reads D i ≡ σ i σ T i /2 = k B T i γ −1 and thus depends linearly on temperature T i .The external force F(x) yields a T i -independent drift −Ax = γ −1 F(x), where A is generally non-symmetric but confining, i.e. the eigenvalues of A have positive real parts.Thus, x t is ergodic but irreversible with zero-mean Gaussian NESS density and thus depends linearly on the temperature T i .Eq. (3) implies for all T i the decomposition into reversible [77], where α T i = −α i is an antisymmetric matrix [78].
We focus on temperature quenches-instantaneous changes of the environmental temperature at fixed drift.The thermodynamics of relaxation upon a quench T i → T w is fully specified by D i t , as the adiabatic entropy production (housekeeping heat divided by T w ) [73] merely embodies the cost of maintaining the NESS [79] and thus need not be considered.Therefore, TED temperatures T h,c correspond to D h 0 = D c 0 and are equal to those of a reversible system at the same T w [57].
Since the initial condition is a zero-mean Gaussian with Σ w i (0) = Σ s,i , the probability density is Gaussian for all times with Σ w i (t) where ⟨•⟩ w i denotes the average over all paths x t at temperature T w evolving from p i s (x).Note that Σ s,i = T i Σ s,w /T w [see Eq. ( 3)].Introducing δ Ti ≡ T i /T w − 1, the generalized excess free energy reads (see [68]) where we introduced the d × d matrix which via Eq.( 5) fully describes relaxation dynamics.As a paradigmatic example for such processes we consider a harmonically confined Rouse polymer with N beads experiencing hydrodynamic interactions [80,81] and internal friction [82][83][84][85] subject to a shear flow, which was investigated experimentally in [86][87][88][89][90][91][92][93][94].For a representative configuration of the NESS ensemble, see Fig. 1a.One may also consider colloidal particles in the presence of non-conservative optical forces [95].The effect of these forces is included in the 3N × 3N drift matrix A and 3N × 3N noise amplitude σ i [68].Evaluating D i t for the heating and cooling processes upon quenches from TED temperatures T h and T c we find D c t < D h t for all t > 0. That is, heating is faster than cooling (the red line in Fig. 1b is at all times below the blue line).This agrees with the relaxation asymmetry predicted [57] and experimentally verified [56] in reversible systems, and provokes the question if this holds for any linear driving.
Systematics of breaking detailed balance.-Wenow systematically assess the influence of non-equilibrium drifts on relaxation upon a temperature quench.As shown above, any linear drift A for i = c, w, h decomposes as Thus, by choosing any antisymmetric matrix α i we alter the NESS current as well as X(t), but neither Σ s,i nor p s (x).We can thus directly compare an NESS with the corresponding reversible system α i = 0 with the same steady state.Note that such a direct comparison is not given in the example in Fig. 1, since the shear flow alters For ω > ωc the eigenvalues are complex.(e) Angle between the covariance matrices Σ w i (t) and Σs,w.(f) Explanation of the counter-intuitive opposing (effective) rotations at small times during heating from Tc/Tw = 0.1.The change dΣ(t) in Eq. ( 4) starting from the initial Σs,i (black ellipse) for dt = 0.05 split into diffusive (yielding the blue ellipse) and drift along the grey streamlines (yielding orange ellipse) contributions.(g-h) D i t for heating and cooling with and without driving on logarithmiclinear and linear-logarithmic scales.The driven system relaxes faster at large t as predicted from the eigenvalues in (e).Grey lines in (h) show the limiting relaxation rates for long times, e −4r 1 t (dashed line) and e −4ℜ(λ 1 )t (solid line).
Σ s,i as it is not of the form α i Σ −1 s,i with α T i = −α i (see [68] for details about the consistent comparison of equilibrium versus nonequilibrium).
We now consider influence of the non-equilibrium driving.For linear drift the relaxation is governed by the eigenvalues of A [96,97].Since Σ s,i is, by definition, symmetric with positive eigenvalues, we can find a matrix β = β T such that β 2 ≡ Σ −1 s,i [98].Thus, the ma- s,i is diagonalizable with positive eigenvalues [99].Therefore, in the absence of driving A = D i Σ −1 s,i expectedly has strictly positive eigenvalues reflecting a monotonous relaxation to equilibrium.
Once we include driving α w ̸ = 0 in the steady-statepreserving form Eq. ( 7), the spectrum may or may not become complex depending on the detailed form of α w , see e.g.Fig. 2a-d.Complex eigenvectors imply that eigendirections where the drift points "straight" towards 0 cease to exist, see Fig. 2a-c.This happens already at arbitrarily small driving if level sets of p s (x) are (hyper)spherical.If some eigenvalues are on the threshold of becoming complex (branching point ω c in Fig. 2d), A may become non-diagonalizable.In terms of the minimal 2d example in Fig. 2 we have that A is non-diagonalizable when ω = ±ω c (see Fig. 2d).
An interesting consequence of driving is that the different dimensions no longer decouple as they do under detailed balance (see Fig. 2a).This means that the ddimensional Langevin equation ( 2) cannot be decomposed into 1d equations and that rotational dynamics may emerge.In the particular case of temperature quenches we find that driving causes a time-dependent rotation of the level sets of P w i (x, t), see Fig. 2e.In agreement with the opposite sings of T i − T w in Eq. ( 4), these rotations occur in opposite directions during heating and cooling, which is a striking new feature of the relaxation asymmetry.The asymmetry implies that thermal relaxation must not be understood as passing through local equilibria at intermediate (effective) temperatures [1], since this would imply a symmetric relaxation independent of the sign of the temperature quench.Moreover, the rotation in opposite directions emphasizes that heating and cooling here evolve along very distinct pathways in the space of probability distributions (see also [56]).
While the initial rotation during cooling follows the direction of driving, most surprisingly the effective rotations during heating initially oppose the direction of the driving (see Fig. 2e).This effect can be traced to the interplay of ("Trotterized" [100]) diffusion and drift during individual small time increments, see Fig. 2f.During heating for an increment dt diffusion alone propagates the black to the more circular blue ellipse.The subsequent drift along the elliptical streamlines propagates this blue ellipse to the orange ellipse that is, however, effectively rotated in the direction opposite to the drift (for further details see [68]).
Accelerated relaxation.-Beforeproving the relaxation asymmetry we discuss the acceleration of relaxation via driving [63][64][65]67].We therefore focus on the real part of the eigenvalues which determines the relaxation timescales.Upon a change of basis we find A ≡ βAβ −1 = βD i β + βα i β where (βα i β) T = −βα i β.Then, for any complex eigenvalue λ of A with eigenvector v ̸ = 0 we This means that the real parts of the eigenvalues in the presence of driving remain not only positive, as required for the existence of a steady state, but even remain in the interval [µ 1 , µ d ].Thus, Eq. ( 8) states that the smallest real part of eigenvalues of A under driving obeys ℜ(λ 1 ) ≥ µ 1 .Note that ℜ(λ 1 ) typically [101] sets the slowest relaxation rate [96,97].Since ℜ(λ 1 ) increases (or does not decrease) upon driving, the latter typically enhances relaxation on long time scales, as already shown in [67].
Driving also affects the adiabatic entropy production.This effect, however, scales trivially, as the adiabatic entropy production increases with increasing Hence, there is no direct connection between faster relaxation and steady-state dissipation, as the influence of driving on the eigenvalues is specific.For example, the acceleration in d = 2 saturates [see ℜ(λ 1 ) in Fig. 2d].More drastically, multiplying α i by a factor larger than 1 in d = 3 may decrease ℜ(λ 1 ) [67].
We see from Eq. ( 6) that X(t) ∼ e −2ℜ(λ1)t for long times and therefore D i t ∼ e −4ℜ(λ1)t (see [68] and Fig. 2gh).The statement "accelerated relaxation", ℜ(λ 1 ) ≥ µ 1 , means that both, heating and cooling will at long times be faster.In general the difference between heating and cooling upon driving can become larger or smaller than for reversible dynamics with the same Σ s,i , but as we now prove heating is always faster than cooling.
Proof of relaxation asymmetry in driven systems.-Wenow prove the relaxation asymmetry for the dynamics in Eq. (2), i.e. ∆D t ≡ D h t − D c t > 0 for all t > 0. By Eq. ( 6) To prove the asymmetry we must understand the properties of X(t), which is T i -independent.Using the steadystate Lyapunov equation (3) we can rewrite X(t) as where s,w is the driving-reversed version of A as in Eq. ( 7).This form is reminiscent of the dual-reversal symmetry [77,[102][103][104] stating that timereversal in non-equilibrium steady states requires concurrent current reversal.Eq. ( 10) is illustrated in Fig. 3a.The proof again requires to change the basis via β as where we used βA −α β −1 = A T and e − A T t = (e − At ) T .Thus, X(t) is symmetric and hence diagonalizable with )/2t at t = 1 as a function of driving ω (see [68]).For large driving the directions mix, such that the system effectively approaches a circular parabola with stiffness (r1 + r2)/2, which is the real part of eigenvalues in Fig. 2d.real eigenvalues.Since, det e − At = e −tr At , we have det X(t) = e −2tr At ̸ = 0. Therefore, X(t) and thus X(t) have positive eigenvalues x t j > 0, j = 1, . . ., d [99].Although A may have complex eigenvalues or even be non-diagonalizable and exp(−At) may be rotational (see Figs. 2c and 3a), X(t) has a real eigensystem since consecutive rotations in forward and current-reversed directions effectively cancel rotations, see Eq. ( 10) and Fig. 3a.
Using the eigenvalues x t j > 0 we rewrite Eq. ( 9) as If all x t j ∈ (0, 1), the proof for reversible systems [57,61] asserts that ∆D t > 0. It therefore suffices to show that x t j < 1 for all j, which is equivalent to ||X(t)|| < 1, where ||M|| ≡ sup v∈R d \0 ||Mv|| 2 /||v|| 2 and ||v|| 2 = √ v T v are the matrix and Euclidean norm, respectively.Eq. ( 10) does not help in showing this [105]; although eigenvalues of A have positive real parts [see Eq. ( 8)], it may be that ||e −A±αt || > 1 (e.g. the distance to 0 in Fig. 3a increases along the white line).This is possible because the eigenvectors of A are not orthogonal.
We thus change the basis as in Eq. ( 11 Since || X(t)|| = ||X(t)|| this implies x t j < 1 and with Eq. ( 12) completes the proof of ∆D t > 0 for all t > 0.
The proof provides important insight into the thermodynamics of the asymmetry in reversible versus driven systems.Namely, ∆D t in Eq. ( 12) for a driven system at any t is equal to that of any reversible system with drift matrix Â having eigenvalues μi satisfying e −2μj t = x t j .Therefore, at each t the relaxation asymmetry of a driven system is isomorphic to that of an equilibrium system with different geometry (see Fig. 3b for effective stiffness axes of the 2-dimensional parabolic potential), which implies the persistence of the asymmetry.This provokes intriguing questions about the existence of the asymmetry in the presence of time-dependent driving.
Conclusion.-We have proven that overdamped ergodic systems driven by linear drift, conservative or not, for any pair of thermodynamically equidistant temperature quenches warm up faster than they cool down.The relaxation asymmetry [57], which was recently confirmed experimentally [56], therefore persists in driven systems.As the original proof hinged on microscopic reversibility, this finding is surprising and is explained by a non-trivial isomorphism between driven and reversible processes.In the presence of driving a striking new feature of the relaxation asymmetry appears: rotational dynamics emerge with opposite directions during heating and cooling, respectively.This further highlights that small, noisy systems do not relax by passing through local equilibria [1].Moreover, rotations in opposing directions emphasize that heating and cooling evolve along fundamentally distinct pathways [56].An analysis with the framework of "thermal kinematics" [56] will bring even deeper insight.Our results motivate further studies on the existence of the relaxation asymmetry in temporally driven systems [49,[107][108][109][110][111], systems with nonlinear drift [25,27,28,30,112], and in the presence of inertial effects [35].
Supplemental Material for: Asymmetric Thermal Relaxation in Driven Systems: Rotations go Opposite Ways Cai Dieball 1 , Gerrit Wellecke 1,2 and Aljaž Godec 1 1 Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany 2 Present address: Theory of Biological Fluids, Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany In this Supplementary Material we provide further details on model examples, arguments, and calculations presented in the Letter.Besides several technical details, we give the equations and parameters describing the the Rouse chain in confined shear flow and derive and solve equations for the covariance.We extensively elaborate on the rotations in different directions, and address the consistent comparison of equilibrium and non-equilibrium steady states.We conclude with a discussion of the log-norm inequality.In the classical Rouse model (i.e.without hydrodynamic interactions, internal friction, confinement and shear), the time-dependent position of the beads x t is described by the 3N dimensional Langevin equation (denoting the spring stiffness by κ and solvent friction by γ) where the connectivity matrix k is a 3N × 3N matrix that reads (1 3 is the 3d unit matrix and all terms not shown are 0) Note that the temperature dependence is contained in the diffusion constant D i ∝ T i as described in the Letter.Following Ref. [1], we introduce hydrodynamic interactions in the pre-averaging approximation via the 3N × 3N matrix H with 3 × 3 entries for m, n = 1, . . ., N and also include internal friction with friction parameter ξ IF , such that the equation of motion of the chain becomes We now introduce a spatial confinement via a 3d harmonic potential centered at x = 0 and subject the confined chain to a shear flow in the x-y-plane.To account for the confinement, we consider the three-dimensional matrix (r i > 0) and for the shear flow in the x-y-plane Note that by introducing the shear flow S to the system, the microscopic reversibility is lost, i.e., we have an example that is genuinely driven out of equilibrium.
To avoid the special case where shear and confinement are orthogonal, we introduce the rotation matrix R(θ) to rotate the confinement by an angle θ within the x-y−plane, Now consider a 3N dimensional matrix M with R(θ)CR T (θ) + S on the N diagonal 3 × 3 blocks to impose the confinement and shear on each bead.Then the equation of motion for the confined polymer in shear flow reads This may now be rewritten as Upon identifying we arrive at the desired form dx t = −Ax t dt + σ i dW t as described in the Letter.The parameters used in Fig. 1 are N = 20, r x = 0, r y = 0.1, In Fig. 1a we choose ω = 3 while in Fig. 1b we use ω = 20 to emphasize the differences between the curves.For the chosen parameters, several eigenvalues of the matrix A become complex, thus confirming that the Rouse model in the shear flow is an irreversible process.the probability density) effectively rotates in the counterclockwise direction while the individual particles on average follow the rotational drift in the clockwise direction, see Fig. S1c.The emergence of this counterintuitive opposing rotation is explained in Fig. 2f in the Letter.To repeat this, during a Trotterized time-increment the diffusion propagates the initial covariance ellipse to a more circular (less eccentric) one.Next, note that the rotational drift is not a perfect circulation, but instead driving along elliptical contour-lines plus the driving into the center due to the confining (conservative) potential.This clockwise elliptical rotational driving applied to the ellipse (previously "rounded" during the diffusion Trotter-increment) leads to the counter-clockwise rotation directly by following the streamlines of the drift (see Fig. 2f in the Letter).
To elaborate on these rotations consider Fig. S2.Ellipses in Fig. S2 and those shown below are the covariance ellipses, while ellipses in Fig. S1 and Fig. 2f in the Letter correspond to standard-deviation ellipses (i.e., square roots of covariance ellipses).In Fig. S2a we recall the opposite rotation during heating and cooling.As in Fig. S1 we then focus on the heating, where the initial rotation is in the counterintuitive direction; see Fig. S2b.To illustrate the explanation given in Fig. 2f in the Letter, we show that this rotation similarly emerges if we start in a circular initial condition (see Fig. S2c).
If we instead consider a circular driving with circular steady-state density (see Fig. S2d) all rotations emerge in the (intuitive) clockwise direction, which shows that the elliptical (i.e.non-circular) component of the circular driving is a key factor in this phenomenon.In Fig. S3 we further illustrate the relation between the direction of rotation and the shape of covariance ellipses.As explained above and in Fig. 2f in the Letter, a more circular (less eccentric) ellipse [sign(3−ratio)=1] leads to a surprising counter-clockwise rotation [sign(angle-change)=1].The overlap of the curves in Fig. S3c,d corroborates this explanation.Small deviations between the curves emerge since the heuristic explanation only applies to ellipses with angle(t) = 0.In Fig. S4 we repeat the presentation of Fig. S3 for a case where the eigenvalues of the drift matrix are real (see Fig. 2d in the Letter for ω < ω c ).We observe that (opposite) rotational motions also occur for the case of real eigenvalues, which illustrates that (effective) rotational motions do not only emerge for complex eigenvalues.A difference with respect to Fig. S3 is that the angles do not cross 0 such that the explanation for the overlaps in Fig. S4c,d only applies at t = 0. FIG.S4.As in Fig. S3 but for ω = 0.9ωc, i.e., eigenvalues of the drift matrix A are real (see Fig. 2d in the Letter).

Relevance and generality of the observation of counterintuitive rotations
So far we only investigated the origin of the counterintuitive rotations in the two-dimensional example.However, since such counterintuitive rotations already occur in this linear, low-dimensional example, it is to be expected that such motions also occur for more general driven systems.In particular, if two-dimensional subspaces are described by the example above one immediately has this rotation in the subspace of the more general dynamics.Generally, one expects opposite rotations during heating and cooling (and therefore one of the two has to rotate opposite to the driving) due to the difference in sign of Σ s,i − Σ s,w = (T i /T w − 1)Σ s,w for i = h, c in Eq. (S13) [Eq.(4) in the Letter] as pointed out above.
The relevance of this observation is twofold.On the one hand, it further emphasizes the asymmetry between heating and cooling, and that the process does not pass through locally equilibrated states the system cannot be described by a time-dependent temperature).On the other hand, it also relevant for general relaxation phenomena, i.e. beyond thermal relaxation.For example, imagine one observes the part of the relaxation in Fig. S1a,b for t ∈ [0, 0.1].If one only observes the apparent counterclockwise rotation the probability one would never guess that the underlying driving is actually in the clockwise direction.Therefore, awareness of this counterintuitive phenomenon might prove useful to avoid false conclusions; and a deep understanding of this phenomenon helps to arrive at correct conclusions.

V. CONSISTENT COMPARISON OF EQUILIBRIUM AND NON-EQUILIBRIUM STEADY STATES
We here discuss under which circumstances we consider a comparison of equilibrium (EQ) and non-equilibrium steady states (NESS), or of different NESS, to be consistent.
In short, we consider a comparison to be consistent if tuning the driving strength does not change the steady-state density.Before we explain this in detail, we want to stress that a consistent comparison is by no means required for the statement of the thermal relaxation asymmetry to be valid, since this statement is proven for any NESS with linear drift in the Letter.Therefore, we were able to chose the physical example of a Rouse chain in a shear flow to illustrate the relaxation asymmetry in Fig. 1 in the Letter (which in fact does not represent a consistent comparison).The consistent comparison is, however, necessary for the statement of "accelerated relaxation" since this statement compares the relaxation speed towards an NESS with the relaxation speed in the corresponding passive system relaxing into an equilibrium steady state.
In the Letter, we use Eq. ( 3), i.e.Eq. (S12), to obtain the decomposition A = (D i + α i )Σ −1 s,i [Eq.( 7) in the Letter] with α T i = −α i for the linear drift matrix A. Note that here D i , α i , Σ s,i ∝ T i all increase linearly with temperature but the product A involving Σ −1 s,i ∝ T −1 i is temperature independent.Any A from this decomposition fulfills Eq. (3) in the Letter, i.e.Eq. (S12), with the given Σ s,i , and in turn any A implying a steady-state covariance Σ s,i via Eq.( 3) in the Letter can be decomposed with this Σ s,i according to A = (D i + α i )Σ −1 s,i .The advantage of the latter form is that it allows to systematically compare NESS dynamics (or in the special case reversible dynamics) dx t = −Ax t dt + σdW t with different A that possess different driving strengths but the same steady-state density.This comparison is performed by tuning the parameter α i (reversible systems are obtained by setting α i = 0) for a given Σ s,i , which then yields A via A = (D i + α i )Σ −1 s,i [Eq.( 7) in the Letter].We consider such a comparison to be consistent, in contrast to a comparison where tuning the irreversible driving alters Σ s,i and thus the steady-state density.
An example for a driving that does not yield a consistent comparison is the shear flow in Fig. 1 in the Letter and in Eqs.(S1)-(S9).We discuss this comparison detail now.For simplicity we consider a single particle N = 1 in the x-y plane to the confining potential and shear flow [see Eqs.The shear flow ω ̸ = 0 renders the dynamics irreversible.However, since now it is not of the form αΣ −1 s with α T = −α as in Eq. ( 7) in the Letter, the steady-state covariance for ω ̸ = 0 will no longer be given by Eq. (S17), i.e. the steadystate Lyapunov equation [see Eq. (3) in the Letter or Eq.(S12)] for A with ω ̸ = 0 will give rise to another steady-state different from Eq. (S17) which corresponds to ω = 0. Therefore, comparing systems with different ω will generally not be consistent [opposed a comparing systems with different α i in Eq. ( 7) in the Letter].
We illustrate this inconsistent comparison by three different examples.Choosing the parameters r x = 1, r y = 0.1, ω = 3, θ = −10 • as in Fig. 1a in the Letter, the eigenvalues of A are 0.55 ± 0.51i, i.e. compared to ω = 0 with eigenvalues r x,y the statement of faster relaxation as quantified in Eq. ( 8) in the Letter does still hold true, even though the proof does not apply here (see also Fig. 1b in the Letter where the curves with the shear flow decay faster at long times).However, if one instead takes ω = 0.5, θ = 10 • the eigenvalues of A are 1.08 and 0.02 i.e. the limiting relaxation is slower compared to the reversible system since 0.02 < r x,y .Thus the statement of faster relaxation does not apply since the effect of the shear flow on the steady state is too large.Even more extreme is the case ω = 3, θ = 10 • where the eigenvalues are 1.365 and −0.265where the negative eigenvalue implies that the shear flow destroyed the confining potential in the sense that the resulting drift no longer corresponds to a confined process.This

FIG. 1 .
FIG. 1.(a) Configuration of a harmonically confined (color gradient) Rouse polymer with N = 20 beads in 3d with hydrodynamic interactions and internal friction subject to a shear flow (arrows) in the x-y-plane drawn from the NESS with covariance Σs,w (see [68] for parameters); a projection onto the x-y-plane is shown.(b) The corresponding free energy difference D i t in Eq. (5) during heating from Tc (red) and cooling from T h (blue) with (solid lines) and without (dashed lines) irreversible shear flow.The shear changes D i t , but the thermal relaxation asymmetry D c t < D h t for t > 0 remains valid.Inset: Temperatures Ti before the quench are chosen thermodynamically equidistant, i.e.D c 0 = D h 0 .

FIG. 2 .
FIG. 2. (a-c) Steady-state density p w s (x) (color gradient) and streamlines of the drift field −Ax for a 2d motion in Eq. (2) with σw = √ 21 and drift matrix A with elements Ajj = rj with r1 = 1, r2 = 3, A jk = (−1) j ωr k for j, k ∈ {1, 2}, with ω in units of ωc ≡ |r2 − r1| /2 √ r1r2.Real eigendirections (yellow) only exist for ω ≤ ωc.(d) Real and imaginary parts of eigenvalues of A as a function of ω.At ω = ωc the eigenvalues coincide and eigendirections (yellow lines in b,c) merge, i.e.A is not diagonalizable.For ω > ωc the eigenvalues are complex.(e) Angle between the covariance matrices Σ w i (t) and Σs,w.(f) Explanation of the counter-intuitive opposing (effective) rotations at small times during heating from Tc/Tw = 0.1.The change dΣ(t) in Eq. (4) starting from the initial Σs,i (black ellipse) for dt = 0.05 split into diffusive (yielding the blue ellipse) and drift along the grey streamlines (yielding orange ellipse) contributions.(g-h) D i t for heating and cooling with and without driving on logarithmiclinear and linear-logarithmic scales.The driven system relaxes faster at large t as predicted from the eigenvalues in (e).Grey lines in (h) show the limiting relaxation rates for long times, e −4r 1 t (dashed line) and e −4ℜ(λ 1 )t (solid line).

CONTENTSI. 7 VIII. Effective stiffness 7 IX. Log-norm inequality 8 References 8 I
Rouse polymer with hydrodynamic interactions and internal friction in confined shear flow 1 II.Lyapunov equation and time-dependent covariance 3 III.Generalized excess free energy during heating and cooling 3 IV.Effective rotations opposing the direction of drift 3 Relevance and generality of the observation of counterintuitive rotations 5 V. Consistent comparison of equilibrium and non-equilibrium steady states 6 VI.Adiabatic entropy production 7 VII.Long-time scaling of the Kullback-Leibler divergence .ROUSE POLYMER WITH HYDRODYNAMIC INTERACTIONS AND INTERNAL FRICTION IN CONFINED SHEAR FLOW In Fig. 1 in the Letter we consider the motivating example of a polymer chain with N = 20 beads in 3d space represented by the Rouse model with internal friction and hydrodynamic interactions in shear flow.That is, we assume that the beads are connected by harmonic springs with zero rest length [Eq.(S1)] and additionally interact via hydrodynamic interactions [Eq.(S3)] and experience internal friction [Eq.(S4)].The chain is confined in a parabolic potential and is subject to a shear flow [Eqs.(S8)-(S9)].We now describe the interactions and evolution equations individually, with increasing complexity.

FIG
FIG. S2. (a,b) Covariance ellipses from Eq. (S13) for heating and cooling for the process as in Fig. S1 and Fig. 2 in the Letter.(c) As in (b) but with initial condition with r1 = r2 = 2.(d) As in (b) but with process (but not initial condition) defined with r1 = r2 = 2.
FIG. S3.(a) Same as in 2e in the Letter.(b) Ratio of the axes of the covariance ellipse.Values below 3 reflect more circular (less eccentric) ellipses compared to the initial condition and the steady state.(c,d) Direction of rotation (clockwise rotation is +1) and indicator of shape (±1 means more/less round, i.e., less/more eccentric) for heating (c) and cooling (d).