Celebratio Mathematica

In the cen­tur­ies since the dis­cov­ery of New­ton equa­tion, the quest to solve the many-body prob­lem has been one of the most per­sist­ent en­deav­ours of math­em­at­ics and phys­ics. Al­though pro­gress was made in ap­prox­im­at­ing it when the num­ber of particles is small, a solu­tion for large num­bers of particles in any use­ful form seems simply im­possible. The fun­da­ment­al ob­ser­va­tion of Boltzmann was that the typ­ic­al be­ha­vi­or for clas­sic­al Hamilto­ni­an sys­tems in equi­lib­ri­um is gov­erned by en­semble av­er­ages (Gibbs states, in today’s lan­guage). This avoided the dif­fi­culty of dir­ectly solv­ing the New­ton equa­tions by pos­tu­lat­ing stat­ist­ic­al en­sembles, and led to mod­ern stat­ist­ic­al phys­ics and er­god­ic the­ory. Boltzmann’s for­mu­la­tion con­cerned sys­tems in equi­lib­ri­um; in oth­er words, be­ha­vi­or of sys­tems as the time ap­proaches in­fin­ity.

At the oth­er end is the kin­et­ic the­ory for short-time be­ha­vi­or. Clas­sic­al dy­nam­ics are ex­actly solv­able when there is no in­ter­ac­tion among particles, that is, in the case of free dy­nam­ics. For short time, clas­sic­al dy­nam­ics can be un­der­stood by sup­ple­ment­ing the free dy­nam­ics with col­li­sions. The fun­da­ment­al ob­ser­va­tion of kin­et­ic the­ory — the ideal­iz­a­tion of the col­li­sion pro­cesses — is again due to Boltzmann in his cel­eb­rated work on the Boltzmann equa­tion.

For sys­tems neither in equi­lib­ri­um nor near free dy­nam­ics (that is, for time scales too short for equi­lib­ri­um the­ory but too long for kin­et­ic the­ory) the most use­ful de­scrip­tions are still the clas­sic­al mac­ro­scop­ic equa­tions, for ex­ample, the Euler and Navi­er–Stokes equa­tions. These are con­tinuum for­mu­la­tions of con­ser­va­tion of mass and mo­mentum, and also con­tain some phe­nomen­o­lo­gic­al con­cepts such as vis­cos­ity. They are equa­tions for mac­ro­scop­ic quant­it­ies such as dens­ity, ve­lo­city (mo­mentum) and en­ergy, while the Boltzmann equa­tion is an equa­tion for the prob­ab­il­ity dens­ity of find­ing a particle at a fixed po­s­i­tion and ve­lo­city. The clas­sic­al Hamilto­ni­an plays no act­ive role in either for­mu­la­tion, and all the mi­cro­scop­ic ef­fects are sum­mar­ized by the vis­cos­ity in the Navi­er–Stokes equa­tions, or the col­li­sion op­er­at­or in the Boltzmann equa­tion. However, the cent­ral the­or­et­ic­al ques­tion, that is, un­der­stand­ing the con­nec­tion between large particle sys­tems and their con­tinuum ap­prox­im­a­tions, re­mained un­solved and is still one of the fun­da­ment­al ques­tions in nonequi­lib­ri­um stat­ist­ic­al phys­ics.

Since clas­sic­al dy­nam­ics of large sys­tems are all but im­possible to solve, a more feas­ible goal is to re­place the clas­sic­al dy­nam­ics with stochast­ic dy­nam­ics. From the 1960s to early 1980s, tre­mend­ous ef­fort was made by Dobrush­in, Le­bowitz, Spohn, Pre­sutti, Spitzer, Lig­gett, and oth­ers to un­der­stand large stochast­ic particle sys­tems. A key fo­cus was to de­rive rig­or­ously the clas­sic­al phe­nomen­o­lo­gic­al equa­tions from the in­ter­act­ing particle sys­tems in suit­able scal­ing lim­its. The meth­ods at the time were based on coup­ling and per­turb­at­ive ar­gu­ments; the sys­tems which can be treated rig­or­ously were re­stric­ted to spe­cial one-di­men­sion­al sys­tems, per­turb­a­tions of the sym­met­ric simple ex­clu­sion or mean-field type in­ter­ac­tions.

To­geth­er with Guo and Papan­icoloau, Varadhan [3] in­tro­duced the first gen­er­al ap­proach, the en­tropy meth­od. The key ideas are the dis­sip­at­ive nature of the en­tropy, and large de­vi­ations. As long as the equi­lib­ri­um meas­ures of the dy­nam­ics are known, and the scal­ing is dif­fus­ive, this ap­proach is very ef­fect­ive and now has been ap­plied to many sys­tems. We will now sketch the ap­proach in [3]. The meth­od is most trans­par­ent in a mod­el (the hy­dro­dynam­ic­al lim­it of this mod­el was first proved un­der some more re­strict­ive as­sump­tions in [e1]) where the particle num­ber is re­placed by a real-val­ued scal­ar field ${\phi }_x\in \mathbb{R}$, $x\in \{1,\cdots ,N\}$, with peri­od­ic bound­ary con­di­tion so that $N+1=1$. These evolve as in­ter­act­ing dif­fu­sions. De­note by ${\mu }_0$ the product meas­ure such that the law of ${\phi }_x$ is giv­en by $e^{-V({\phi }_x)}$. Let $f_t{\mu }_0$ be the dis­tri­bu­tion of the field $\phi$ at the time $t$. The dy­nam­ics of $\phi$ will be giv­en by the evol­u­tion equa­tion for $f_t$, $$\begin{array}{rl}\text{(1)}& & {\partial }_t{f}_t=L{f}_t,& L=\sum _{j=1}^N(\frac{\partial }{\partial {\phi }_j}-\frac{\partial }{\partial {\phi }_{j+1}}{)}^2-\sum _{j=1}^N(V^{\prime }({\phi }_j)-V^{\prime }({\phi }_{j+1}))(\frac{\partial }{\partial {\phi }_j}-\frac{\partial }{\partial {\phi }_{j+1}}).\end{array}$$ This dy­nam­ics is re­vers­ible with re­spect to the in­vari­ant meas­ure ${\mu }_0$, and the Di­rich­let form is giv­en by $$\begin{array}{rl}\text{(2)}& D(f):& =-\int fLfd{\mu }_0& =\sum _{j=1}^N\int (\frac{\partial f}{\partial {\phi }_j}-\frac{\partial f}{\partial {\phi }_{j+1}}{)}^{2}d{\mu }_0\end{array}$$ We res­cale the time dif­fus­ively so that the evol­u­tion equa­tion be­comes $$\begin{array}{}\text{(3)}& {\partial }_t{f}_t={\epsilon }^{-2}L{f}_t,\end{array}$$ where $\epsilon =N^{-1}$ is the scal­ing para­met­er.

The dy­nam­ics $\text{(1)}$ can be writ­ten as a con­ser­va­tion law, $$\begin{array}{rlr}& d{\phi }_i=(w_{i+1}-w_i)dt+d{M}_i,& w_i=N^2{V}^{\prime }({\phi }_i)-N^2{V}^{\prime }({\phi }_{i-1})\end{array}$$ where $M_i$ are mar­tin­gales and $w_i$ is the mi­cro­scop­ic cur­rent. The cur­rent is it­self a gradi­ent, and our dy­nam­ics is form­ally dif­fus­ive, that is, for any smooth test func­tion $J$, $$\begin{array}{}\text{(4)}& d{N}^{-1}\sum _{i}J(\epsilon i){\phi }_i\sim N^{-1}\sum _i{J}^{\prime \prime }(\epsilon i)V^{\prime }({\phi }_i)dt+dM.\end{array}$$ De­note by $\rho (x,t)$ the loc­al av­er­age of $\phi$ around $x=\epsilon i$, $$\begin{array}{}\text{(5)}& \rho (x,t)=\underset{\delta \to 0}{lim}\underset{\epsilon \to 0}{lim}\frac{1}{2\epsilon \delta +1}\sum _{|j-i|\le \delta {\epsilon }^{-1}}{\phi }_j.\end{array}$$ Let $p(\lambda )$ de­note the pres­sure $$\begin{array}{}\text{(6)}& p(\lambda )=\log {\int }_{\mathbb{R}}d\phi e^{\lambda \phi -V(\phi )},\end{array}$$ and $h(m)$ de­note the free en­ergy, that is, the Le­gendre trans­form of $p$. It is not hard to check that the mar­tin­gale term in $\text{(4)}$ van­ishes in the lim­it, and hence that the main task for es­tab­lish­ing the hy­dro­dynam­ic­al lim­it of the form $${\partial }_t\rho ={\partial }_{xx}{h}^{\prime }(\rho )$$ is to prove that we can re­place $V^{\prime }({\phi }_i)$ in $\text{(4)}$ by $h^{\prime }(\rho (x,t))$, in the sense of a law of large num­bers with re­spect to the dis­tri­bu­tion $f_t$ sat­is­fy­ing $\text{(3)}$.

Con­sider the loc­al Gibbs meas­ure with a chem­ic­al po­ten­tial $\lambda$, $$d{\mu }_{\lambda }({\phi }_1,\cdots ,{\phi }_N)=\exp [\sum _j\lambda (\epsilon j){\phi }_j]d{\mu }_0$$ where the chem­ic­al po­ten­tial $\lambda$ is al­lowed to de­pend on the site slowly. If $f_t$ is a loc­al Gibbs state, then cer­tainly we can re­place $V^{\prime }({\phi }_i)$ in $\text{(4)}$ by $h^{\prime }(\rho (x,t))$ in the sense of a law of large num­bers. The key ob­ser­va­tion of [3] is to con­sider the evol­u­tion of the en­tropy, $$\begin{array}{rl}{\partial }_{t}S(f_t)& =-D(\sqrt{f_t}),\\ S(f)& =\int f\log fd{\mu }_0.\end{array}$$ For a typ­ic­al sys­tem of $N$ vari­ables giv­en by a dens­ity $f$ with re­spect to ${\mu }_0$, the en­tropy $S(f)$ is of or­der $N$. This im­plies that $$\begin{array}{}\text{(7)}& {\int }_0^{t}D(\sqrt{f_s})ds\le CN\end{array}$$ This in­form­a­tion alone is suf­fi­cient to es­tab­lish that the solu­tion of $\text{(3)}$ is close enough to a loc­al Gibbs state that the law of large num­bers con­tin­ues to hold.

Sys­tems where the cur­rent is it­self a gradi­ent of some oth­er func­tion are known as gradi­ent sys­tems. For such sys­tems, [3] provides a gen­er­al frame­work to es­tab­lish the hy­dro­dynam­ic­al lim­it. However, many sys­tems are of non­gradi­ent type. A simple il­lus­trat­ive ex­ample is to modi­fy the dy­nam­ics $\text{(1)}$ so that the gen­er­at­or $L$ be­comes the sym­met­ric op­er­at­or with Di­rich­let form $$\begin{array}{rl}\text{(8)}& D(f):& =-\int fLfd{\mu }_0& =\sum _{j=1}^N\int a({\phi }_j,{\phi }_{j+1})(\frac{\partial f}{\partial {\phi }_j}-\frac{\partial f}{\partial {\phi }_{j+1}}{)}^{2}d{\mu }_0.\end{array}$$ The cur­rent can be com­puted eas­ily, and is giv­en (up to scale factors) by $$\begin{array}{}\text{(9)}& w_j=-\frac{\partial a({\phi }_j,{\phi }_{j+1})}{\partial {\phi }_j}+\frac{\partial a({\phi }_j,{\phi }_{j+1})}{\partial {\phi }_{j+1}}+a({\phi }_j,{\phi }_{j+1})[{\phi }_1-{\phi }_{j+1}].\end{array}$$ To es­tab­lish the hy­dro­dynam­ic­al lim­it, it is now re­quired to prove that, when $x=\epsilon j$, in some sense $w_j$ can be re­placed by $$\begin{array}{}\text{(10)}& w_j=D(\rho (x,t))[{\phi }_{j+1}-{\phi }_j]\end{array}$$ for some func­tion $D$, which will be the dif­fu­sion coef­fi­cient of the hy­dro­dynam­ic­al equa­tion. The key ob­ser­va­tion of Varadhan’s work on non­gradi­ent sys­tems is that $$\begin{array}{}\text{(11)}& w_j=D(\rho (x,t))[{\phi }_{j+1}-{\phi }_j]+LF\end{array}$$ where $F$ is some loc­al func­tion of ${\phi }_j$ and $L$ is the gen­er­at­or giv­en by $\text{(8)}$. The idea is that func­tions of the type $LF$ rep­res­ent in­co­her­ent rap­id fluc­tu­ations which van­ish over the long time scale of the hy­dro­dynam­ic­al lim­it. This fluc­tu­ation is in­deed in the sys­tem, and the hy­dro­dynam­ic­al lim­it can be es­tab­lished only if we prop­erly ac­count for its ef­fect.

The sense in which $\text{(11)}$ holds is the $H_{-1}$ sense, cor­res­pond­ing to the van­ish­ing of the vari­ance in the cent­ral lim­it the­or­em for the cor­res­pond­ing ad­dit­ive func­tion­al. This goes back to Varadhan’s earli­er work on tagged particles [2]. The prob­lem of prov­ing the con­ver­gence of tagged particles to ap­pro­pri­ate dif­fu­sions is some­what com­ple­ment­ary to the hy­dro­dynam­ic­al lim­it. Varadhan in­tro­duced the mar­tin­gale meth­od in this con­text so that the idea of view­ing the sys­tem from the point of view of the particle can be im­ple­men­ted. These ideas have had broad in­flu­ence not only in hy­dro­dynam­ic­al lim­its, but also in ho­mo­gen­iz­a­tion the­ory and for ran­dom walk in ran­dom en­vir­on­ment. In fact, there is even a more ex­pli­cit con­nec­tion between the tagged particle prob­lems and the non­gradi­ent sys­tems. Sup­pose that one gives each particle one of $m$ dif­fer­ent la­bels, and watches the evol­u­tion of the $m$ dif­fer­ent dens­it­ies in the hy­dro­dynam­ic­al lim­it. The cor­res­pond­ing particle sys­tems are usu­ally of non­gradi­ent form [e5] as long as $m\ge 2$. This is a weak form of tag­ging, and the large-$m$ lim­it of this sys­tem is a (weak) way to keep track of in­di­vidu­al particles. It can be proved, via non­gradi­ent sys­tem meth­ods, that each spe­cies of particles evolves ac­cord­ing to a dif­fu­sion equa­tion and, thus, the hy­dro­dynam­ic­al lim­it of tagged particles in nonequi­lib­ri­um is es­tab­lished [23]. The ad­vant­age of this ap­proach is that it can be done in nonequi­lib­ri­um, identi­fy­ing the col­lect­ive drift im­posed by the flow of the bulk to­wards equi­lib­ri­um. However, it is strictly speak­ing not the be­ha­vi­or of a single tagged particle, but the av­er­age be­ha­vi­or of tagged particles with van­ish­ing dens­ity.

The equa­tion $\text{(11)}$ is quite dif­fi­cult to solve as it in­volves the full gen­er­at­or $L$. In or­der to solve it, Varadhan de­veloped a meth­od which can be viewed as an in­fin­ite-di­men­sion­al ver­sion of Hodge the­ory. This is a deep the­ory, and we shall only at­tempt to con­vey some of its fla­vor here. First note that, be­cause of the en­tropy bound $\text{(7)}$, one only has to solve $\text{(11)}$ in equi­lib­ri­um. So, the dif­fu­sion coef­fi­cient can be treated as a con­stant. For sim­pli­city take $j=0$. The cur­rent $w_0$ has the prop­erty that $$\begin{array}{}\text{(12)}& \int w_{0}d{\mu }_{\lambda }=0\end{array}$$ for any Gibbs state with con­stant chem­ic­al po­ten­tial $\lambda$. The space of func­tions with this prop­erty cor­res­ponds to a space of closed forms. A sub­space of ex­act forms cor­res­ponds to the fluc­tu­ation terms $LF$. The deep res­ult is that the ex­act forms are of codi­men­sion one in the space of closed forms with or­tho­gon­al com­ple­ment cor­res­pond­ing to ${\phi }_{j+1}-{\phi }_j$; this solves $\text{(11)}$. This ap­proach, as it stands, is based on the in­teg­ra­tion-by-parts nature of $L$ and ap­plies only to re­vers­ible dy­nam­ics. It is pos­sible to for­mu­late it also for non­re­vers­ible dy­nam­ics, and the form­al ana­logy between this equa­tion and the Hodge the­ory can be strengthened [22].

The two fun­da­ment­al pa­pers [3], [14] of Varadhan ushered in an era of hy­dro­dynam­ic­al lim­its based on the idea of en­tropy. The de­vel­op­ments fol­low­ing these two pa­pers are as­ton­ish­ing, and we shall only men­tion a few. The ap­proach of [3] was suc­cess­fully ap­plied to many sys­tems, in­clud­ing in­ter­act­ing Browni­an mo­tions [9], in­ter­act­ing Orn­stein–Uh­len­beck pro­cesses [8] and Gin­zburg–Land­au mod­els [e2]. The in­ter­act­ing Browni­an mo­tions and in­ter­act­ing Orn­stein–Uh­len­beck pro­cesses are con­tinuum sys­tems with no lat­tice struc­ture. The hy­dro­dynam­ic­al lim­it for the Gin­zburg–Land­au mod­els was proved for all tem­per­at­ures, in­clud­ing the phase trans­ition re­gion — a re­mark­able res­ult. Fur­ther­more, the ap­proach of [3] was suc­cess­fully ap­plied to kin­et­ic scal­ing, and led to the de­riv­a­tion of the Boltzmann equa­tion from stochast­ic particle sys­tems [e9]. The idea that the solu­tion of $\text{(3)}$ is heur­ist­ic­ally a loc­al Gibbs states goes back many dec­ades, but the es­tim­ates ob­tained in [3] are in fact strong enough to prove it. It is ob­served in [e4] that one can by­pass many tech­nic­al dif­fi­culties in [3] and prove dir­ectly that the loc­al Gibbs states are in fact an ap­prox­im­ate solu­tion to $\text{(3)}$ in the sense of re­l­at­ive en­tropy. The as­sump­tions needed in this ap­proach are (a) some er­god­ic prop­er­ties of the dy­nam­ics, and (b) smooth­ness of solu­tions to the hy­dro­dynam­ic­al equa­tions. This meth­od is more re­strict­ive than [3] for dif­fus­ive sys­tems, but it es­sen­tially re­lies only on the iden­ti­fic­a­tion of the in­vari­ant meas­ures of the dy­nam­ics, and it ap­plies also to hy­per­bol­ic sys­tems be­fore the form­a­tion of shocks. It was ad­ap­ted in [13] to de­rive the clas­sic­al Euler equa­tion from Hamilto­ni­an sys­tems with van­ish­ing noise. This is the most sig­ni­fic­ant ad­vance since Mor­rey stated this prob­lem in the 1960s. Once the hy­per­bol­ic equa­tions de­vel­op shocks, a very dif­fer­ent meth­od is needed, see [e3], [e8] for ref­er­ences and re­lated res­ults.

Varadhan’s work on non­gradi­ent sys­tems re­quires a spec­tral gap of or­der ${\ell }^{-2}$ for the sys­tem in a box of side length $\ell$. This in­spired work on the es­tim­ates of spec­tral gaps of con­ser­vat­ive dy­nam­ics, and it led to the de­vel­op­ment of mar­tin­gale meth­ods for es­tim­at­ing spec­tral gap for con­ser­vat­ive dy­nam­ics. Us­ing this spec­tral es­tim­ate, Varadhan and his coau­thor [22] es­tab­lished the hy­dro­dynam­ic­al lim­it of lat­tice gas in the high-tem­per­at­ure phase.

The idea that the cur­rent can be de­com­posed in­to a dis­sip­at­ive term ${\phi }_j-{\phi }_{j+1}$ and a fluc­tu­ation term $LF$ is a deep idea, and is really a rig­or­ous state­ment of the so-called fluc­tu­ation-dis­sip­a­tion the­or­em from phys­ics. In a sense, the in­sight that this equa­tion is fun­da­ment­al to the hy­dro­dynam­ic­al lim­it is at least as sig­ni­fic­ant as the solu­tion of this equa­tion for the spe­cif­ic mod­el con­sidered in [14]. Al­though the fluc­tu­ation-dis­sip­a­tion equa­tion was solved in [14] only for re­vers­ible dy­nam­ics, it was real­ized that one can de­vel­op a meth­od to solve this equa­tion for non­re­vers­ible dy­nam­ics, provided that the spa­tial di­men­sion is lar­ger than two [e6]. This led to the de­riv­a­tion of the in­com­press­ible Navi­er–Stokes (INS) equa­tions from stochast­ic lat­tice gases for di­men­sion $d=3$ [e7]. The res­ult ob­tained in [e7] is very strong; it iden­ti­fies the large de­vi­ation rate that the hy­dro­dynam­ic­al equa­tion is not a Leray solu­tion, and does not as­sume that the INS equa­tions have clas­sic­al solu­tions. The phys­ic­al sig­ni­fic­ance is the fol­low­ing: The first prin­ciples equa­tion gov­ern­ing a clas­sic­al flu­id is the New­ton equa­tion, which is time re­vers­ible and has no dis­sip­a­tion. The INS equa­tions pos­sess vis­cos­ity and are time ir­re­vers­ible. There­fore, a de­riv­a­tion of the INS equa­tions from clas­sic­al mech­an­ics would have to an­swer the fun­da­ment­al ques­tion re­lat­ing to the ori­gin of dis­sip­a­tion and the break­ing of time re­vers­ib­il­ity in clas­sic­al dy­nam­ics. Al­though the un­der­ly­ing dy­nam­ics in [e7] is stochast­ic, it was proved that the vis­cos­ity in the INS equa­tions was strictly lar­ger than the ori­gin­al vis­cos­ity of the un­der­ly­ing stochast­ic dy­nam­ics. In oth­er words, the de­term­in­ist­ic part of the dy­nam­ics makes a non­trivi­al con­tri­bu­tion to the vis­cos­ity. We re­mark that the con­di­tion $d=3$ is crit­ic­al. For di­men­sion $d=2$, it was proved that the hy­dro­dynam­ic­al lim­it equa­tions for such lat­tice gas mod­els are not the INS equa­tions. In­deed, even the dif­fus­ive scal­ing is in­cor­rect, and there are log­ar­ithmic cor­rec­tions. Al­though these works do not an­swer dir­ectly the fun­da­ment­al ques­tion re­gard­ing the de­riv­a­tion of the in­com­press­ible Navi­er–Stokes equa­tions from the clas­sic­al dy­nam­ics, it is the first time we un­der­stand the gen­er­a­tion of the vis­cos­ity from many particle dy­nam­ics. These de­vel­op­ments are largely at­trib­uted to Varadhan’s in­sight of the im­port­ance of the fluc­tu­ation-dis­sip­a­tion equa­tions $\text{(11)}$.