Thermalization of random motion in weakly confining potentials

Thermalization of random motion in w eakly confining pot en tials Piotr Garbaczews k i and Vladimir Stephanovic h Institute of Physics, University of Op ole, 45-052 Op ole, Pol and (Dated: Jan uary 24, 2021) W e show that in w eakly confining conserv ative force fields, a sub class of d iffusion-type (Smolu- cho wski) processes, admits a family of ”heavy-tailed” non-Gaussian equilibrium probability density functions ( pd fs), with none or a fin ite number of moments. These pdfs, in the standard Gibb s- Boltzmann form, can b e also in ferred directly from an extremum principle, set for Sh an n on en tropy under a constraint that th e mean v alue of the force p otential has b een a priori prescrib ed. That enforces the corresponding Lagrange multiplier to pla y the role of inv erse t emperature. W eak con- fining prop erties of th e p otentials are manifes ted in a thermody namical peculiarity that thermal equilibria can b e approac h ed only in a b ounded temp erature interv al 0 ≤ T < T max = 2 ǫ 0 /k B , where ǫ 0 sets an energy scale. F or T ≥ T max no equilibrium p df exists. P A CS num b ers: 05.40.Jc, 02.50.Ey , 05.20.-y , 05.10.Gg W e depart from a folk lore assumption that a ther mo- dynamical system ha s a state characterized by a pr oba- bilit y densit y , [1]. That amoun ts to studying (random) dynamical systems in ter ms of time-dep endent probabil- it y density functions (p dfs) and discov ering whether and how the system may a pproach a state of thermo dynami- cal equilibrium. Lo ca lization pro per ties of p dfs, b oth far from and at equilibrium, can b e quan tified in terms of Shannon en tropy . Admissible dynamical equilibria can be inferr ed from v ar ious v ar iational principles. A t this p oint we in voke a class ification of maximum ent ropy pr inciples (MEP) a s giv en in Ref. [2]. Let us lo ok for pdfs that derive from so-called first inv erse MEP: given a p df ρ ( x ), c ho ose a n appr opriate set of co nstraints such that ρ ( x ) is obtained if Shannon measur e of entropy is maximized (strictly sp eaking , extr emized) sub ject to those cons tr aints. Namely if a s ystem e v olves in a potential V ( x ) (at the moment, we co nsider a co ordinate x to b e dimensionles s), we can int ro duce the following functional L { ρ ( x ) } = − λ Z ∞ −∞ V ( x ) ρ ( x ) dx − Z ∞ −∞ ρ ( x ) ln[ ρ ( x )] dx, (1) where (without limitation of g e nerality) w e cons ider the one-dimensional case. The first term comprises the mean v alue of a potential. A constant λ is (as y et physi- cally unidentified) Lagr ange multiplier, which tak es car e of aforement ioned constra ints. The s e c ond term stands for Shannon en tropy of a contin uo us (dimensionless) p df ρ ( x ). An extremum of the functional L { ρ ( x ) } can b e found by mea ns of standard v ariationa l arg umen ts and gives rise to the following g eneral form of an ex tr emizing pdf ρ ∗ ( x ) ρ ∗ ( x ) = C exp( − λV ( x )) , (2) which, if rega rded a s the Gibbs-Boltzmann p df, implies that the parameter λ ca n b e int erpreted as inv er s e tem- per ature, λ = ( k B T ) − 1 ( k B is Boltzmann co nstant) , at which a state of equilibrium (asymptotic p df ) is reached by a random dynamical system in a co nfining po ten tial V ( x ). A deceivingly simple question has been p osed in chap. 8.2.4 of Ref. [2]. Having dimensionless loga r ithmic p o- ten tial V ( x ) = ln(1 + x 2 ), o ne should b egin with ev alu- ating a mean v alue U = hV i ≡ R ∞ −∞ V ( x ) ρ ( x ) dx . N ext one needs to show that only if this p articular value is prescrib ed in the a bove MEP pro cedure, Cauch y distri- bution will ultimately ar ise. Additionally , one should answer what k ind of distribution would a rise if any other po sitive exp ectation v a lue is chosen. The a nswer prov es not to b e that straig h tforward and we shall ana ly ze this issue b elow. T o handle the pr oblem w e a dmit a ll p dfs ρ ( x ) for which the mean v alue h ln(1 + x 2 ) i exists, i.e. takes whatever finite p ositive v alue. Then, we adopt the previous v ari- ational procedur e w ith the use of Lag range multipliers. This pro cedure s hows that what we extremize is not the (Shannon) en tro p y itself, but a functional F with a clear thermo dynamic connotatio n (Helmholtz free energ y ana - log, [5]): Φ( x ) = α V ( x ) + ln ρ ( x ) → F = h Φ i = α hV i − S ( ρ ) . (3) Here S ( ρ ) = −h ln ρ i and α is a La g range mult iplier. F rom now on we, we consider a parameter multiplier in the form α = ǫ 0 / ( k B T ), where ǫ 0 is characteristic en- ergy scale of a system. Note that Eq. (3), pr ovides a dimensionless version of a familia r formula F = U − T S , relating the Helmholtz free energy F , in ternal energ y U and entrop y S o f a random dynamica l system. The extremum condition for F δ F ( ρ ) δ ρ = 0 (4) yields an extremizing p df in the form ρ α ( x ) = 1 Z α (1 + x 2 ) − α (5) 2 - 2 - 1 0 1 2 0 . 0 0 . 1 0 . 2 0 . 3 1 ( x ) 0 . 8 ( x ) 0 . 7 ( x ) 0 . 6 ( x ) 0 . 5 1 ( x ) V ( x) , ( x) X FIG. 1: ln(1 + x 2 ) against ρ α ( x ) with 1 / 2 < α ≤ 1. provided the normalization factor Z α = R ∞ −∞ (1 + x 2 ) − α dx exists. It turns out that the in tegr al ca n be ev aluated explicitly in terms o f Γ - functions Z α = √ π Γ( α − 1 / 2) Γ( α ) , α > 1 / 2 , (6) so that we can in princ iple deduce a numerical v a lue of the parameter α , by re s orting to o ur a s sumption that the mean v alue hV i α has actua lly b een fixe d. W e obser ve that the extremizing p dfs ρ α (5) a ppea r to constitute a one-para meter fa mily of p dfs, named b y us Cauch y family , [3, 4]. It is seen from E q. (6) that the ex- tremizing functions (5) are nor ma lizable only for α > 1 / 2 as at α → 1 / 2 w e have Z α → ∞ . W e illustr ate this b e- havior in Fig. 1, which shows that as α → 1 / 2 the whole pdf ρ α ( x ) b e comes identically equa l to zer o. T o deduce the v a lue s of above dimensionless in verse temperatur e α , we need an ex plicit expre ssion for the mean v alue U α = hV i α = Γ( α ) √ π Γ( α − 1 / 2) Z ∞ −∞ ln(1 + x 2 ) (1 + x 2 ) α dx. (7) It turns out that it can b e giv en in terms of digamma function ψ ( x ) = d (ln Γ) / dx [6]. W e get U α = − 2 π sin(2 π α ) + ψ (1 − α ) − ψ  3 2 − α  , α > 1 2 . (8) The dep endence o f U α on α is rep orted in Fig. 2, where it is s een that this function is divergent at α = 1 / 2 (see also b elow) a nd decays mono tonously at large α . This decay is conro lled b y an asymptotic e xpansion U α ≈ 1 2 α + 3 8 α 2 + 1 4 α 3 + ..., (9) which is also shown on Fig. 2. It is seen that decay of U α at large α ob eys the in verse pow er law. It follows 0 1 2 3 4 5 0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0 3 S ( ) 1 2 FIG. 2: U α (curve 1) and its asymptotic expansion at large α (curve 2). from Fig. 2 that the expansio n (9) giv es a v ery go o d approximation of U α for α > 3. W e note that, a pparently , Eq. (8) in volv es a nother divergence pro blems, if we c ho os e integer α . This o b- stacle can be circ um vented by tra nsforming E q. (8) to an equiv alent form that has no (effectiv ely r emov able) divergencies. Namely , we get U α = − π tan π α + ψ ( α ) − ψ  3 2 − α  (10) and the tangent contribution v anis hes for integer α . On the other hand, this ex pression shows tha t the divergence of U α at α → 1 / 2 or iginates fr om the first term in (10), as ψ functions hav e finite v a lues at this p oint. Near α = 1 / 2 the first term of Eq. (10) div erges as ( α − 1 / 2) − 1 , whic h is clear ly seen in Fig. 2. The prec eding discussion shows that weak confinement prop erties of logarithmic p otential manifest themselv es in the fact that the corre s po nding (Cauch y family) equilib- rium p dfs exist o nly for the semi- infinite range of (dimen- sionless) in verse tempe r atures α ∈ (1 / 2 , + ∞ ). Note that for strongly co nfining p otentials (like e.g. V ( x ) ∼ x 2 or x 4 ) this is not the case as it follows fr om Eq. (2) that the normalizing integral C − 1 = R ∞ −∞ exp[ − λV ( x )] dx is con- vergen t at an y λ , so that the whole temp erature ra nge [0 , ∞ ) is allowed. Let us turn to a quan titativ e discussion of an impa ct of weak co nfinement pro per ties of the a bove log arithmic po ten tials o n thermo dyna mical prop erties of a random system. T o this end, we s ho uld focus our atten tion on the thermo dyna mic meaning of the (or iginally La grange m ultiplier) para meter α and its s e mi-bo unded range of v ariability (1 / 2 , + ∞ ). This is also rela ted to ab ov e pro b- lem o f explicit calculation o f α for a sp ecific p df. With an explicit express ion for Cauch y family p dfs (5) 3 0 1 2 3 4 5 0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0 3 S ( ) 1 2 FIG. 3: S α for Cauch y family (curve 1) and its asymp totic expansion at larg e α (curve 2) . S G α for Gaussian fami ly is also sho wn (curve 3). in hands, we readily ev a luate Shannon en tro p y to obtain S α = − Z ∞ −∞ ρ α ( x ) ln ρ α ( x ) dx = ln Z α + α U α . (11) Then, the (as yet dimens io nless) Helmholtz free energ y F α reads F α = α U α − S α ≡ − ln Z α , (12) with α b eing the dimensionless inverse temper a ture, c.f. Ref. [5]. W e note, that in view of the divergence of Z α , bo th the Shannon entropy and the Helmholtz free energy (likewise U α ) cease to exist at α = 1 / 2 . W e plot S α as a function of α in Fig.3. It is seen that entrop y monoto nously decays for α > 1 / 2 and fo r lar ger v alues of α . An asymptotic expansion o f the en tropy shows loga rithmic plus in verse power signatures S α ≈ 1 2  1 − ln α π  + 3 4 α + 3 8 α 2 + ... (13) These ser ies are shown along with the entropy in Fig. 3 . As α grows, the n umber of moments of respec tive p dfs increases. That allows to exp ect that an ”almo st Gaus- sian” b e havior should be displayed by α ≫ 1 members o f Cauch y fa mily (5) . This is indeed the case. T o this end, let us reca ll that the normalized Gaussian function with zero mean and v a r iance σ 2 = 1 / 2 α reads ρ G α ( x ) = r α π e − αx 2 ≡ 1 Z G α e − αx 2 . (14) The co rresp onding Shannon entropy giv es exactly the first term of E q. (13) S G α = 1 2 ln π e α . (15) The plot of S G α is repor ted in Fig.3 together with S α for Ca uc h y family . This equality suggests that at large α Ca uch y family p dfs (5 ) tra nsit to Gaussian one (14). This transition can b e re v ealed if we rew r ite Cauch y fam- ily p dfs (5) in the ex p onential form ρ α ( x ) = exp(ln ρ α ) where ln ρ α = − ln Z α − α ln(1 + x 2 ). It can b e shown that at lar ge α the ”tails” o f p df do not make substantial contribution so that only s mall x play a ro le. This means that in limiting case of large α w e can expand ln(1 + x 2 ) at small x . Then, employing series expansions Z − 1 α →∞ ≈ r α π − 3 8 √ π α − 7 128 α √ π α − ... (16) we arrive a t ρ α ( x ) ≈  r α π − 3 8 √ π α ...  exp  − αx 2 + α x 4 2 ...  (17) whose leading order gives exac tly the no rmalized Gaus- sian (14). Accordingly , c.f. Fig. 3, a n inequality α > 5 sets a regime where a fair ly go o d approximation of the Cauch y family p dfs b y Gaussian is v alid. Note tha t ul- timately , in the α → ∞ limit, b oth families of functions (sequentially) approach the Dirac delta function(al). Let us now show that the v ariatio nal principle (1) ex- plicitly iden tifies an equilibr ium so lution of the F o kker- Planck equation for standard Smoluc howski diffusion pro cesses. T o address our thermaliza tion issue co rrectly , we now use dimensional units. The F okker - P lanck e q ua- tion that drives an initial pro babilit y dens ity ρ ( x, t = 0) to its final (equilibrium) fo r m ρ ( x, t → ∞ ) rea ds ∂ t ρ = D ∆ ρ − ∇ · ( b ( x ) ρ ) . (18) W e still refer (altho ug h without substan tial limitation of generality) to the 1D ca se, so that ∇ ≡ ∂ /∂ x . Here, the drift field b ( x ) is time-independent and conserv ative, b ( x ) = −∇ V ( x ) / ( mγ ) ( V ( x ) is a p otential, while m is a mass and γ is a recipro ca l r e laxation time of a system). W e keep in mind that ρ and bρ v anish at spatial infinities or other int egration int erv al b orde r s. If Einstein fluctuation-dissipation r elation D = k B T /mγ holds, the eq ua tion (18) can b e identically rewritten in the form ∂ t ρ = ∇ [ ρ ∇ Ψ] / ( mγ ), where Ψ = V + k B T ln ρ (19) whose mean v a lue is indeed the Helmholtz free energy of random motion F ≡ h Ψ i = U − T S. (20) Here the (Gibbs) entropy rea ds S = k B S , while an inter- nal energy is U = h V i . In v iew of as sumed bounda r y re- strictions at spatial infinities, we hav e ˙ F = − ( mγ )  v 2  ≤ 0. Hence, F decreases as a function of time towards its minim um F ∗ , or remains c onstant. 4 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 t =0 . 0 1 0 . 1 1 * 1 ( x ) x 1 d i f ( x ) ( a ) - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 x t =0 . 0 1 0 . 1 1 * 2 ( x ) 2 d i f ( x ) ( b ) - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 x 3 d i f ( x ) t =0 . 0 1 0 . 1 1 * 3 ( x ) ( c ) FIG. 4: Time evolution of pdf ’s ρ ( x, t ) for Smolucho wski pro cesses in logarithmic p otential ln ( 1 + x 2 ). The initial ( t = 0) p df is set to b e a Gaussian with height 25 and half-width ∼ 10 − 3 . The first depicted stage of evolution corresponds to t = 0 . 01 . T arget p dfs are the members of Cauc hy family (5) for α = 1 (pan el (a)), 2 (panel (b)) and 3 (panel (c)) respectively . Let us co nsider the stationary (large time a symptotic) regime asso ciated with an inv ariant density ρ ∗ (c.f. [7] for an extended discussion of that issue). Then, ∂ t ρ = 0 a nd we hav e ∇ Ψ [ ρ ∗ ] = C ρ ∗ ( C is arbitr a ry constant) which yields ρ ∗ = 1 Z exp[ − V /k B T ] . (21) Therefore, at equilibrium: Ψ ∗ = V + k B T ln ρ ∗ = ⇒ h Ψ ∗ i = − k B T ln Z ≡ F ∗ , (22) to b e co mpared with Eq. (12). Here, the partition func- tion equals Z = R exp( − V /k B T ) dx , provided that the int egral is conv ergent. Since Z = exp( − F ∗ /k B T ) we hav e recas t ρ ∗ ( x ) in the familiar Gibbs-Boltzmann for m ρ ∗ = exp[( F ∗ − V ) /k B T ]. On physical grounds, V ( x ) carries dimensions of en- ergy . Therefor e to esta blis h a physically justifiable ther- mo dynamic picture of Smolucho wski diffusion pro cesses, relaxing to Ca uc h y family p dfs in their lar ge time asymp- totics, we nee d to assume that logarithmic p otentials V ( x ) = ln(1 + x 2 ) a re dimensiona lly sca le d to the form V ( x ) = ǫ 0 V where ǫ 0 is an arbitrary constant with phys- ical dimensions of energy . By emplo y ing (1 /ǫ 0 ) V ( x ) = ln(1 + x 2 ), w e can re- cast previous v aria tional argumen ts (MEP proc edure) in terms of dimensiona l thermo dyna mica l functions. Namely , in view of k B T Φ( x ) = V ( x ) + k B T ln ρ ( x ) w e hav e an obvious transformation of Eqs . (3), (5) and (12) int o Eqs. (2 0),(21) a nd (22) resp ectively with k B T Φ( x ) = Ψ( x ). The ab ov e dimensional arg uments tell us that in weakly confining logarithmic po ten tials, Cauch y family of pdfs can b e rega rded as a one-pa r ameter fa mily of e quilibrium p dfs, where the reservoir temp erature T en- ters thro ugh the exponent α . P ro ceeding in this vein, we note that ǫ 0 should be reg a rded a s a characteristic energy (energy s cale) of the consider ed rando m system. W e observe that α → ∞ corresp onds to T → 0 i.e. a maximal lo calization (Dirac delta limit) of the corre - sp onding p df. In pa r allel S ( α → ∞ ≡ T → 0) → 0 and an analo gy with the Nernst the or em is establishe d . The opp osite limiting cas e α → 1 / 2 look s interesting as well. Na mely , w e hav e S ( α → 1 / 2) → ∞ . T o grasp the meaning of this limiting re gime, we rew r ite α = 1 / 2 in the form k B T = 2 ǫ 0 . Accordingly , the temp erature scale, within which our s y stem ma y at all be s e t at ther- mal eq uilibr ium, is b ounded: 0 < k B T < 2 ǫ 0 . F o r tem- per atures exceeding the upper b ound T max = 2 ǫ 0 /k B no thermal e qu ilibrium is p oss ible in the presence of (w eakly , i.e. weaker then, e.g ., V ( x ) ∼ x 2 ) confining lo garithmic po ten tials V ( x ) = ǫ 0 ln(1 + x 2 ). The ca se of α = 1 i.e. k B T = ǫ 0 corres p onds to Cauch y density . Let us finally a dd that in Ref. [3], w e ha ve analyzed diffusion pro ce s ses in logarithmic p otentials, with a fo cus on tempo ral relaxa tion patterns of the pr o cess to wards asymptotic p dfs from the Cauchy family . F or clarity of presentation (dynamical interpolatio n scenario s b etw e e n initial Gaussian and equilibrium Cauch y-type pdfs do not seem to have ever b een considered in the literature), in Fig. 4 w e plot v ar ious stages of the diffusiv e F okker - Planck dyna mics fo r pro cesses that a ll hav e been started from a narrow Gauss ian. It turns out that the the resul- tant (lar ge time asymptotic) equilibrium p dfs a re mem- ber s of Cauch y family (5), labeled res pectively by α = 1 , 2 and 3. [1] M. C. Mack ey , Time’s ar r ow: The origins of thermo dy- namic b ehavior , S pringer-V erlag, Berlin, 1992 [2] J. N. Kapur and H. K. Kesa v an, Entr opy O ptimization Principles with Applic ations , Academic Pres s, Boston, 1992 [3] P . Garbaczewski, V. Stephan ovic h and D. K¸ edzierski, Non-Gaussian targets an d (ab)normal patterns of relax- ation in random motion, arXiv:1004.0127 , (2010) [4] P . Garbaczewski and V. Step hano vich, Phys. R ev. E 80 , 031113, (2009) [5] P . 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