Weak Galilean Invariance As a Selection Principle for Coarse

Downloaded by guest on September 27, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1717292115 a Cairoli Andrea for models principle diffusive selection coarse-grained a as invariance Galilean Weak omi sal nnw.Wiefrnra ifso u to due an diffusion of positions (MSD) normal with particles displacement of for mean-square ensemble While the motion unknown. precise Brownian whose usually interactions, is multiparticle form complex stochasticity to where due phenomena, arises transport anomalous describe are variants their (2–4). and sciences equations the throughout such free- vast of of degrees applications relevant microscopic The the on of dom. noise effects and using, friction coarse-grained level as the interactions diffusion mesoscopic capture Fokker–Planck a to or on equations equations Langevin rather Newtonian stochastic with but level e.g., motion, microscopic a of and on science equa- equations in modeled Newton’s systems not complex of are of form nature variety large functional that a the However, constraints on tion. the imposes a in naturally plays manifest it GI role, fully models, be constitutive can deterministic fundamental evolution microscopic simultaneous dynamical pre- by between whose that characterized distances systems transformations For and (1). affine intervals events time general both in serve transformations of are Galilean equations These to respect con- the (GTs). with interacting their frames invariant all are prin- inertial including stituents, i.e., latter different systems, The closed in of motion. motion that rectilinear is either states uniform forces) exhibits ciple external or of absence rest freely the a at in where systems (i.e., coordinate particle are moving former The (1). (GI) ance C coarse-graining processes stochastic data. a experimental before theoretical against models a validation stochastic provide consistent physically they select as to systems, principle cor- for biological the relevant of particularly derive modeling are the we results Our which except description. for invariant terms, rect walk these random in well- continuous-time invariant Several the are invariance.” processes Galilean stochastic stochastic “weak general known call by satisfied we be which to models, differ- have in that systems frames characterizing rules inertial of equations. ent set the a stochastic during to leads broken deriving analysis Our is when invariance procedure Galilean coarse-graining how Brownian show generating we model starting motion, Hamiltonian Here, Kac–Zwanzig frames. the inertial different from a in no of model consistency seemingly physical stochastic is the given priori there a assess violated, to then principle alternative is invariance forces. exter- Galilean by stochastic and and Since described friction as internal often indistinguishably interactions complex are nal integrating systems world models real coarse-grained Galilean However, dynam- by rules. the related on ical constraints frames strong inertial imposing different thus systems transformations, in closed of same motion invari- the of equations Galilean remain the of that principle prescribes It famous ance. the fundamental formulating this addressed by first 2017) question dif- 2, Galilei in October change frames? review system reference for a (received ferent of 2018 description 16, mathematical April the approved does and How MA, Cambridge, University, Harvard Weitz, A. David by Edited Kingdom United 4NS, E1 London London, of University eateto iegneig meilCleeLno,Lno W A,Uie igo;and Kingdom; United 2AZ, SW7 London London, College Imperial Bioengineering, of Department oreganddfuiemdl r atclryrlvn to relevant particularly are models diffusive Coarse-grained ocpso nrilrfrnefae n aieninvari- Galilean and related frames intimately reference two inertial the upon of built concepts is mechanics lassical | rcinlcalculus fractional a,b | anrKlages Rainer , aieninvariance Galilean b | n dinBaule Adrian and , X nmlu transport anomalous (t ) ttime at t rw linearly grows | b,1 1073/pnas.1717292115/-/DCSupplemental hti fe mrcs u olmtdsml sizes. sample data limited experimental to with due imprecise comparison often the is that only rule different remains between fundamental of distinguish it consistency no To physical models priori. the is a verify models there to stochastic used such far be so could that Unfortunately available 25). effects (5, 24, observables memory various spatiotemporal 7, of statistics on power-law rely non-Gaussian suggested which and been has literature, models the diffusive phenomenological of in mostly wealth a on fact, formulated In grounds. prin- usually micro- first are from the and such derived ciples typically for of be of models details cannot because theoretical processes the anomalous experiments, However, assessing in 23). interactions in (22, scopic difficulties granules intrinsic insulin the and lipid and of of transport intracellular cerevisiae the in in Saccharomyces and e.g., (17); observed, fluctuations protein is in structural molec- (16); diffusion transport a ribonucleoprotein anomalous on messenger Here, neuronal processes (15). cellular level of ular characteristic established ubiquitous was it a (11– Recently, motion as (14). movement biological human Likewise, for even and found (10). 13) later based (CTRW) was 9 walk diffusion and random 8 anomalous refs. continuous-time in the described theoretically on first semiconductors was amorphous in that in diffusion (5–7) transport molecular pro- charge plasmas, physical and in of nanopores, transport range particle wide like a cesses for experimentally observed been hsatcecnan uprigifrainoln at online information supporting contains article This 1 the under Published Submission. Direct PNAS a is article This paper. interest. the of wrote conflict A.B. no and research; declare R.K., authors performed A.C., The A.C. and data; research; analyzed designed A.B. and A.B. R.K., and A.C., R.K., A.C., contributions: Author uini clsnnierywith nonlinearly scales it fusion limit, long-time the in owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To u aet aif ekGlla nainefrphysical for alone invariance data Galilean with weak agreement cho- satisfy be the consistency. to cannot on have general based but results in Our arbitrarily models models. sen stochastic diffusive these that by highlight satisfied invariance”— be Galilean to “weak need that as show rules—denoted we frames, alternative forces, different in stochastic the procedure studying By coarse-graining and invariance. Galilean friction violate intrinsically which as requires indistinguishably usually inter- systems actions microscopic complex world integrating How- under real models frames. of coarse-grained change of description inertial not the motion different do ever, of to freedom equations transformations of the Galilean degrees systems microscopic It closed mechanics. the for classical of that cornerstone states a is invariance Galilean Significance shrci coli Escherichia NSlicense. PNAS b colo ahmtclSine,QenMary Queen Sciences, Mathematical of School el 1,2) nufdmcopee (21), microspheres engulfed 20), (19, cells hX iohnra(8,crmsmlloci chromosomal (18), mitochondria 2 ∼ i . t β β with nmlu yaishas dynamics Anomalous =1. 6 www.pnas.org/lookup/suppl/doi:10. β NSLts Articles Latest PNAS 1 = o nmlu dif- anomalous for , | f6 of 1

PHYSICS ∂ X Here, we show that GI can provide precisely such a constitu- x˙i (t) = vi (t), mi v˙i (t) = − U (xi (t), xj (t)). [3] ∂xi tive principle. Even though the fundamental role of GI seemingly imomentum for simplicity let it be the (N + 1)th, is a tagged (tracer) particle conservation, which makes it unsuitable to simulate hydrody- of mass mN +1 = M that interacts with the remaining particles namic phenomena (32). Curing this deﬁciency led to novel of equal mass mj = m via a harmonic potential of coupling 2 GI algorithms, most notably dissipative particle dynamics, now strength mωj , thus deﬁning the environment as a heat bath; widely used to simulate soft matter systems and simple liquids PN 2 2 i.e., U (X , xj ) = mω [X (t) − xj (t)] /2. Conversely, inter- (33–35). j =1 j actions between different bath particles are switched off. This The basic setup of our problem is represented in Fig. 1: Here is a Galilean invariant version of the classical Kac–Zwanzig S and Se are two inertial reference frames, where S is the lab- model (36), whose relevance has been recently addressed (37). oratory frame at rest while Se is moving with uniform velocity v0 Denoting by (X (t), V (t)) and (xj (t), vj (t)), j = 1, ..., N the with respect to S. The GTs connecting the coordinates in the two position and velocity variables of the tracer and heat bath frames are given by particles, respectively, in the frame S, their Hamilton’s equa- ¨ PN 2 tions become M X (t) = mωj [xj (t) − X (t)] and mx¨j (t) = x = x − v0t, v = v − v0, [1] j =1 e e 2 −mωj [xj (t) − X (t)]. These equations specify the time evolution where, for simplicity, we focus on the one-dimensional case. of all N + 1 particles of the system (arrows in the box in Fig. 1A) Eq. 1 is the phase space version of the classical GTs assum- in S once the initial conditions are prescribed, which we take as ing an absolute time (1). A classical system of N + 1 interacting (X (0), V (0)) = (0, 0) and (xj (0), vj (0)) = (xj 0, vj 0), without loss particles is described by the Hamiltonian function of generality. The great advantage of this model is that the effective dynamics for the tracer can be derived by integrating out the X mi 2 X H (x1, v1; ... ; xN +1, vN +1) = vi (t) + U (xi (t), xj (t)), bath degrees of freedom. This yields (36) 2 i i

B1 P(x,t) C1 STOCHASTIC STATISTICAL t1 COARSE- ANALYSIS t2 GRAINING t3 x

TRANSFORMATION RULES v0

STOCHASTIC STATISTICAL COARSE- ANALYSIS GRAINING ?? A B2 C2

Fig. 1. Pictorial representation of the setup: A system of N heat bath particles (black) and one tracer (red) is observed from two different reference frames S and Se. While S is at rest, Se is moving with velocity υ0 with respect to S. We consider three different levels of description of the original system: (A) The microscopic system of N + 1 particles is described by deterministic equations of motion leading to trajectories fully speciﬁed by the initial conditions. (B1) Alternatively, one can provide a stochastic coarse-grained model of the tracer dynamics in terms of effective dissipative friction forces and random collisions with the N bath particles (arrowed spheres), which account for their original microscopic interactions with the probe. (C1) Finally, the system can be studied in terms of its position and velocity statistics, whose distributions are determined by either experimental measurements or a prescribed stochastic model. While the relationship between the dynamical evolutions in S and S˜ for A is speciﬁed by Galilean transformations of the position and velocity degrees of freedom, here we derive the corresponding relationships for B1 ↔ B2 and C1 ↔ C2, yielding what we call weak Galilean invariance.

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1717292115 Cairoli et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 nrilfae a erltdt ahohrdrcl.Including directly. other each to related be different in can velocity frames and position inertial for (PDFs) probabil- functions the also density Consequently, ity 1). two section the Appendix, in (SI conditions Eqs. frames initial solving different explicitly the for by accounting shown correctly be can which ity, (X processes ( position–velocity resulting the that imply unchanged. term noise the leaving Eq. while from only directly obtained be could 4 which term, drift additional ntodfeetieta rms eseta Eq. that see We frames. inertial different two in Eqs. than ified statistic different a acquires tracer the of in equation coarse-grained deterministic the Overall, Eq. in in of those dynamics the to exact, is related everything are which particles, S bath Eq. the from of seen conditions be can As admfreta a e esnlsotarfrnefaeand frame Eq. to reference according a because out GI, singles breaks inevitably se thus per way is that graining force Eq. coarse random for stochastic directly the that possible such not invariant frame not is in (LE) Eq. equation (38). holds relation k system the of perature in librium of of distribution statistics the microscopic The original probe. the their with for interactions accounting collisions particles, random bath and origi- the forces effectively with friction then dissipative tracer both Eq. the from force of nates dynamics deterministic the the level, of Langevin specifying instead by force description random coarse-grained this simplify microscopic original the the of freedom of of particles. degrees GI the bath out the projecting after maintains even dynamics thus tracer the Eq. using Eq. call i.e., frame, we If particles. bath the Eq. but arl tal. et Cairoli e X B e ypromn To h oriae fisdtriitcpart deterministic its of coordinates the on GT a performing by h rnfrainrlso h tcatceutoso motion of equations stochastic the of rules transformation The o eiigsohsi agvndnmc h etse sto is step next the dynamics Langevin stochastic deriving For by S ξ T e , (t V Ω(| e ste utEq. just then is 4 ξ x e = ) r eae i T vni h rsneo stochastic- of presence the in even GT, a via related are ) and j 1 i h qiiru supinin assumption equilibrium the via 0 M t 6 ξ 1 oEqs. to = ξ e (t h eemnsi fetv qaino oinfor motion of equation effective deterministic The 7. − scagddet h T fteiiilvlcte of velocities initial the of GTs the to due changed is (t X S e ¨ 9 x = ) h eoiydsrbto sMxela ttetem- the at Maxwellian is distribution velocity the , + ) j (t t hnbt ersn h aemcocpcdynamics microscopic same the represent both then 0 2 , |) = ) 6 X j v e = v =1 N 4–6. j 3) osqety h fluctuation–dissipation the Consequently, (36). in 0 S 0 − − x X j = rcal,tento ftemlequilibrium thermal of notion the Crucially, . ω j =1 N ∼ 4 0 Ω(t Z Z j , v Ω 0 0 v in aibe,tetonie r eae by related are noises two the variables, v ω j T t t j 0 j 0 j = ) sucagdudrtetransformation, the under unchanged is Ω(t Ω(t 0 ∼ implying , − 4 sin(ω sin(ω suigta h etbt sa equi- at is bath heat the that Assuming . variables 6, hndfie eeaie Langevin generalized a defines then v X j 0 − − ξ =1 N e pcfigtepoete fthe of properties the Specifying 8. ξ and h os emi h transformed the in term noise the t t j j ω t 0 0 t eed xlctyo h initial the on explicitly depends ) )( + ) = ) j 2 X ξ e ˙ X e cos(ω X oee,atrhvn spec- having after However, . hξ ˙ e S (t e X ξ j (t e 0 0, = (0) (t =1 N 0 (t olwb pligteGTs the applying by follow )dt 0 )i + ) + ) ω j t 0 0 = j 2 ), + S v x v 0 j 0 ξ Eq. , 0 ξ )dt e V (t Z e (t and cos(ω 0 ) 0 = (0) ) t 0 r pcfidby specified are + Ω(t 9 4 hξ 9 ξ 7 j and ssilvalid. still is (t otisan contains (t t 0 −v h noise the )dt ). 1 nthe On 6. ), ξ )ξ , (t 0 while 9, V 0 (t Since . . ) and ) 2 )i sa as [6] [5] [7] [9] [8] = ξ ˜ h oiincodntsas coordinates position the aefrudrapddnmc h D rnfrainrule transformation PDF the dynamics underdamped for have eto is different nection by in to denoted defined according on variables bath independent now fluc- the heat [from over transform same is Fourier–Laplace frames the inertial of both in tuations value expected the since hw otasomvaaG nteridpnetvrals(SI variables independent their on 2 GT section a Appendix, via transform to of shown equations evolution The space are Fourier–Laplace in results respective the dynamics o rcinlBona oin cldBona oin and motion, Brownian scaled Eqs. motion, FP with Brownian or agreement Langevin fractional given in For the solutions to GT yields a representations description applying all i.e., CTRW, GI; the weak from exhibit apart Remarkably, 2 ). tion property of validity the in only ii presented demonstrate we is simplicity overview for where An 51). 10, (5, (42–44), LE fractional fractional as the such L (41–43), superdiffu- literature, motion the and Brownian in scaled sub- used and commonly both are generating that sion models stochastic other first-principles theoretical purely a from consistency perspective. physical models the the stochastic assess can beyond to GI of criterion weak quantities important that an propose statistical as we determina- serve (11), of data ultimately accurate experimental comparison process with the MSD the stochastic While anomalous on 40). underlying 24, relies an procedure 7, of coarse-graining (5, tion rigorous are available similarly processes not a of is which variety large for in a used However, diffusion regime. anomalous anomalous the modeling in preserved are properties Ω Eq. equation. in equation FP the case satisfy 4) motion stochastic of their equations while frames, different. the inertial are equivalents latter, all coarse-grained in the identical satisfying strictly are systems In microscopic conventional from GI. rules GI in weak these ones distinguish the to of with compared shift in unchanged a PDFs remain corresponding the from respectively, variables, Apart position and GI: what weak yield coarse-grained call counterparts for statistical we their rules and dynamics transformation specific stochastic Galilean any procedure. for coarse-graining three shown a These following be hand to PDFs at principle model ( iii) in stochastic and needs and variables, nontrivial independent is their Eqs. on in trans- as GT transform also a equations (ii) via Klein–Kramers consequently, form GT only; and a processes (FP) via velocity Fokker–Planck transform and motion position their as of uniquely on equations frames Stochastic Galilean (i) different follows: all in dynamics Eq. stochastic Nevertheless, GI. violates inherently tion ´ v ihs(54) L (45–47), flights evy dtiso h acltosaedsusdin discussed are calculations the of (details nfc,w eie h aiiyo u ojcuefrseveral for conjecture our of validity the verified we fact, In (Eq. LE generalized the by described processes all Clearly, descrip- coarse-grained stochastic a that shown have we far So oe-a enli ie(9,wihhglgt htthese that highlights which (39), time in kernel power-law a P (x hc nldsnra ifsv rcse.I this In processes. diffusive normal includes which i–iii, P , v (k , 4 t , = ) p lomdl nmlu ifso foeue for uses one if diffusion anomalous models also = = , = λ) P D hδ ). e P 10 δ (x (x (k (x e and , − ´ v ak 2,4–0,adteCTRW the and 48–50), (24, walks evy − S −ipv − λ e v = ) X stewl-nw advection–diffusion well-known the is X 0 e h aiiyo h properties the of validity The 11. t 0 (t (t P X , P P e ˙ e v ))δ ) (x (t (k (k − − = ) , (v , , v v v p λ 0 (x 0 , , − , P t − t V λ t )δ ) , V (x ), − v iv (t and (v , NSLts Articles Latest PNAS (t , 0 ) al S1, Table Appendix, SI ikv t t k v ) − ))i = ) and 0 ). → 0 P V e o overdamped For ). or S 7 (x P (k e S sec- Appendix, SI (t ntrso its of terms In . hrceie the characterizes X e ˙ (x ti important is It . v , , ) 0 t (t p − ) t − , = ) λ a lobe also can v o velocity for v 10 0 h con- the )] 0 ) V t E e , and (t | t ) ) f6 of 3 [10] [11] and i–iii 11. we S e

PHYSICS the fractional LE (as a special case of the generalized LE), this and SI Appendix, section 3). Therefore, a simple transformation result can be proved based on the Gaussian nature of the process. of the fractional diffusion equation obtained by arbitrarily adding For Lévy flights it is a direct consequence of the Lévy–Khintchine an advective term v0∂/∂x as for the Gaussian models and Lévy representation of Lévy processes (52). In these examples, the flights (SI Appendix, Table S1) is not correct. Likewise, imple- Langevin dynamics can be expressed in terms of an additive menting GTs directly on the Langevin description of CTRWs in noise process and thus the transformation into frame Se by GT terms of subordination (51, 54, 57) is problematic (shown below). is unproblematic, leading to an advective term v0∂/∂x as for Instead, the correct transformation of Eq. 12 into the frame normal diffusion. Even though such a simple structure does not Se can be derived straightforwardly in Fourier–Laplace space. apply to Lévy walks, surprisingly the same consistency is satis- Without loss of generality, we assume P(x, 0) = δ(x). Thus, its fied, as can be checked by imposing a GT onto the respective FP transform is λP(k, λ) − 1 = −σk 2[λ/Φ(λ)]P(k, λ). Using prop- equation (50) and verifying that the solutions in each frame are erty iii, the GT is then implemented by the variable transfor- related by Eq. 11. The FP equation in Se describes a Lévy walk mation λ → λ + ikv0 and the transformation rule Eq. 11 relat- with asymmetric velocity jumps switching between −v0 + u and ing P, Pe. This immediately leads to a FP equation including −v0 − u, where ±u is the velocity in S, which clearly is physically retardation effects correct. ∂ ∂ We now clarify the situation for the CTRW, a model that has (v0) Pe(x, t) = v0 Pe(x, t) + LDt Pe(x, t), [13] huge applications across all branches of the sciences (5–7, 40). In ∂t ∂x P S the overdamped regime the PDF of a CTRW in the frame (v0) is the solution of the diffusion equation (53, 54) where the operator Dt is the fractional substantial derivative (49, 54, 58) ∂ ∂2 ∂ ∂ Z t P(x, t) = LDt P(x, t), L = σ 2 , [12] (v0) 0 0 0 0 ∂t ∂x Dt Pe(x, t) = − v0 dt K(t − t )Pe(x + v0(t − t ), t ), ∂t ∂x 0 where σ is a generalized diffusion constant and t is a nonlocal [14] D (v ) ∂ t 0 0 0 0 R which has Fourier–Laplace representation Dt Pe(x, t) → (λ + time operator defined as Dt P(x, t) = ∂t 0 dt K (t − t )P(x, t ), which generalizes the Riemann–Liouville fractional differential iv0k)Pe(k, λ)/Φ(λ + iv0k). Setting v0 = 0 recovers Eq. 12. operator to arbitrary waiting-time distributions. The kernel K is To further support our result, we also derive Eq. 13 directly related to the so-called Laplace exponent Φ of the waiting-time in (x, t) space. This requires a careful analysis due to the nonlo- −1 distribution by K (λ) = Φ(λ) (53, 54). Therefore, its Fourier– cal character of the operator Dt . On the one hand, the left-hand 12 Laplace representation is Dt P(x, t) → λP(k, λ)/Φ(λ). In the side (lhs) of Eq. and the time derivative in Dt transform CTRW framework a constant drift can be incorporated by com- with the substitution ∂/∂t → ∂/∂t − v0∂/∂x (chain rule applied plementing the diffusion operator with v0∂/∂x, which would sug- to Eq. 1). On the other hand, recalling the explicit definition ∂ ∂ of a PDF in terms of probability (denoted as ) of events gest that the FP equation in Se is given by Pe = v0 + L t Pe. P ∂t ∂x D (denoted as {·}), the integrand PDF is defined as P(x, t 0) = Alternatively, another time nonlocal FP equation was previously 0 derived, in particular for Φ(λ) = λα (0 < α < 1) corresponding to P({x ≤ Y (t ) ≤ x + dx}), where Y (t) denotes the position of 0 0 0 Lévy stable distributed waiting times, by using the transformation the CTRW. According to property i, Y (t ) becomes Ye (t ) + v0t rule Eq. 11 and performing a Taylor expansion in the Fourier in the comoving frame Se, while the measured position x trans- variable up to the lowest approximation order (5, 55, 56). This forms at the later time t in agreement with the lhs of Eq. 0 0 ∂ ∂ 12; i.e., x → x + v0t. Therefore, P(x, t ) = ({x + v0(t − t ) ≤ procedure leads to the equation ∂t Pe = v0 ∂x Pe+LDt Pe. P 0 0 0 0 However, both equations are not correct representations of Ye (t ) ≤ x + v0(t − t ) + dx}) = Pe(x + v0(t − t ), t ). Note that microscopic dynamics in view of the rules i–iii yielding weak GI. dx is invariant because the shift cancels out. Combining these In fact, the former does not satisfy the general rule Eq. 11 as arguments yields Eq. 13. The fractional substantial derivative in becomes clear by solving it in Fourier–Laplace space. The same Eq. 14 highlights the existence of a space–time coupling, which is true for the latter, whose solutions are even unphysical, as they is absent in the frame S but is naturally required: Let y be the do not satisfy the requirement of positivity of a PDF (Fig. 2A position of the CTRW in S after its last jump occurred at time

0.4 0.5 0.4 A t=1 B t=1 C t=1 0.3 t=2 0.4 t=2 t=2 t=4 t=4 0.3 t=4 0.2 t=6 0.3 t=6 t=6 0.2 ~ ~ P(x,t) P(x,t) 0.1 0.2 P(x,t) 0.1 0 0.1 0.1 0 0 5 0 5 10 15 5 0 5 10 15 5 0 5 10 1 5

∂ ∂ Fig. 2. Position distribution in the comoving frame Se.(A) Propagator of the FP equation ∂t eP = υ0 ∂x eP+LDteP suggested in refs. 5, 55, and 56 instead of Eq. 13. The explicit expression for the propagator is given in SI Appendix, Eq. S47. This function not only violates weak GI, but also exhibits nonphysical negative α values. Here, Φ(λ) = λ with α = 0.5, υ0 = −1, σ = 1, x0 = 0. (B) PDF solution of Eq. 13 (SI Appendix, Eq. S68) showing weak GI. Parameters are the same as ˙ for A. We ﬁnd perfect agreement with Monte Carlo simulations of the Langevin equation Ye(t) = −υ0 + ξ(t) (colored markers). (C) PDF solution of Eq. 13 (SI Appendix, Eq. S68) with K(t) = tα−1/Γ(α) and α analytically continued to yield superdiffusion at α = 1.5. Other parameters are the same as for A. Again, weak GI is observed.

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Weiss EW, Montroll 10. 4 rcmn ,Hfae ,Gie 20)Tesaiglw fhmntravel. human of laws scaling The (2006) cell T of Geisel L, dynamics L Hufnagel D, Anomalous Generalized Brockmann (2012) (2008) 14. al. A et Schwab TH, R, Harris Preuss 13. R, Klages P, Dieterich 12. npyia ieadteeuvln hrceiainof characterization equivalent description the Langevin and the time on physical relies in fact surprising This fusion. ne rn PL2951o h niern n hsclSine Research funding Sciences Physical acknowledges Labora- and gratefully Engineering Council. Mathematical A.B. the of Fellow. London EP/L020955/1 External Grant an the under support. is from financial he for funding where Global tory, acknowledges Research of Naval also Science Exhibition of the Office the R.K. for the under Commission thanks and Royal R.K. London the 1851. of by University granted Fellowship Mary Research Queen by in granted presented Fund derivation the of 3 support technical for ACKNOWLEDGMENTS. chemical, physical, sciences. the biological throughout and principle approaches can selection which modeling data, important guide with an comparison preceding provides models it stochastic for such, conven- As satisfy to GI. expected tional is models representation diffusive microscopic mesoscopic reaching. all far whose constrain term thus to advective are expected hoc is results mod- GI our ad Weak unphysical of an consequences to with The CTRW be lead 2A). the (Fig. easily to by that can exemplified need is rules as and statement els, GI 67) important weak most (66, our on our ignoring suggested (64) But been thermodynamics further. have rela- investigated of hand fluctuation law other celebrated con- second the the lines, the and these generalizing hand fluctuation–dissipation Along one tions of the (66). validity on the understood relations and well GI not between nections far latter so the of are diffusive thermodynamics stochastic fun- anomalous the highlighting though and even (65), systems, normal flow between external similarities the account of damental to of theory modified contribution be recent the to the for have within (64) used thermodynamics work, production stochastic of of entropy details definitions and the the of heat, Consequently, irrespective coupling. particle underlying tracer addi- the the an to on generally field leads in bath flow heat tive representation the frame the comoving a from in representation that suggested Langevin the the all Moreover, that frame. at shows rest the not which models, is CTRW representation form of missing correct class the important providing the them. by for for this missing demonstrated is procedure We coarse-graining even precise analytical superdiffusion, diffusive and a sub- our anomalous though both for consistent on constructed rules be can based these models diffusion using by normal these While derivation, for frames. inertial hold different properties in system prop- stochastic GI same weak the derived have erties we model, Kac–Zwanzig matic 63. sat- ref. PDF in resulting continued cally 2C the Fig. and In 6), t GI. section methods weak functional Appendix, isfies via (SI Eq. theorem 62) still Novikov’s (61, particu- is of In generalization equation (59). a FP functions correlation its multipoint lar, its of means .Kae ,Rdn ,SklvI (2008) IM Sokolov G, Radons R, Klages 7. .ShrH a 17)Sohsi rnpr nadsree oi.I Theory. I. solid. disordered a in transport Stochastic (1973) M Lax H, Scher 8. .ShrH otolE 17)Aoaostasttm iprini mrhu solids. amorphous in dispersion transit-time Anomalous (1975) EW Montroll H, Scher 9. ..gaeul cnwegsfnigudrtePsgaut Research Postgraduate the under funding acknowledges gratefully A.C. . α −1 nsmay sn aienivratvrino h paradig- the of version invariant Galilean a using summary, In hsRvB Rev Phys 7:4491–4502. cations irto fefco D+Tcells. T CD8+ effector of migration migration. cells. 439:462–465. fln ,Faoc 21)Aoaostasoti h rwe ol fbiological of world crowded the in transport Anomalous (2013) T Franosch F, ofling ¨ o for now /Γ(α), i–iii e rgPhys Prog Rep WlyVH enem Germany). Weinheim, (Wiley-VCH, 12:2455–2477. rcNt cdSiUSA Sci Acad Natl Proc htne ob aifidt ossetydsrb the describe consistently to satisfied be to need that 76:046602. .For α). etakA .Cehi o rifldsusosand discussions fruitful for Chechkin V. A. thank We 1 < α < v 0 105:459–463. Nature epo t rpgtrfor propagator its plot we 0 = 2 nmlu rnpr:FudtosadAppli- and Foundations Transport: Anomalous v ak n h oeo hmknsin chemokines of role the and walks evy ´ Eq. Appendix, (SI hsPFwsfis icse in discussed first was PDF this , 486:545–548. hc a edrvdby derived be can which 13, NSLts Articles Latest PNAS ahPhys Math J section Appendix, SI analyti- S68, 6:167. i discloses hsRvB Rev Phys K | (t Nature ξ f6 of 5 = ) by

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