Celebratio Mathematica

S. R. Srinivasa Varadhan

Varadhan's work on hydrodynamical limits

by Jeremy Quastel and Horng Tzer Yau

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{align}\label{dyn} & \partial_{t} f_{t} = L f_{t}, \\ & L = \sum_{j=1}^N \Bigl( \frac {\partial} { \partial \phi_{j}} - \frac {\partial} { \partial \phi_{j+1}} \Bigr)^{2}- \sum_{j=1}^N\Bigl (V^{\prime}( \phi_{j}) -V^{\prime}( \phi_{j+1}) \Bigr) \Bigl ( \frac {\partial} { \partial \phi_{j}} - \frac {\partial} { \partial \phi_{j+1}} \Bigr) . \nonumber \end{align} This dy­nam­ics is re­vers­ible with re­spect to the in­vari­ant meas­ure \( \mu_{0} \), and the Di­rich­let form is giv­en by \begin{align} \label{Di} D(f) : & = -\int f \,L f \, d \mu_{0} \\ & = \sum_{ j=1}^N \int \Bigl( \frac {\partial f} { \partial \phi_{j}} - \frac {\partial f} { \partial \phi_{j+1}} \Bigr)^{2} d \mu_{0} \nonumber \end{align} We res­cale the time dif­fus­ively so that the evol­u­tion equa­tion be­comes \begin{equation}\label{1} \partial_{t} f_{t} = \varepsilon^{-2}L f_{t}, \end{equation} where \( \varepsilon = N^{-1} \) is the scal­ing para­met­er.

The dy­nam­ics \eqref{dyn} can be writ­ten as a con­ser­va­tion law, \begin{align*} & 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{align*} 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{equation}\label{he} d N^{-1}\sum_{i} J(\varepsilon i) \phi_{i} \,\sim\, N^{-1}\sum_{i} J^{\prime\prime}(\varepsilon i) V^{\prime}( \phi_{i}) \,d t + d M . \end{equation} De­note by \( \rho(x, t) \) the loc­al av­er­age of \( \phi \) around \( x = \varepsilon i \), \begin{equation}\label{de} \rho(x, t) = \lim_{\delta \to 0 }\lim_{\varepsilon \to 0} \frac 1 { 2 \varepsilon \delta + 1} \sum_{|j-i| \le \delta \varepsilon^{-1}} \phi_{j} . \end{equation} Let \( p(\lambda) \) de­note the pres­sure \begin{equation}\label{pre} p(\lambda) = \log \int_{\mathbb{R}} d \phi \,e^{\lambda \phi - V(\phi)} , \end{equation} 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 \eqref{he} 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 \eqref{he} by \( h^{\prime}(\rho(x, t)) \), in the sense of a law of large num­bers with re­spect to the dis­tri­bu­tion \( f_{t} \) sat­is­fy­ing \eqref{1}.

Con­sider the loc­al Gibbs meas­ure with a chem­ic­al po­ten­tial \( \lambda \), \[ d \mu_\lambda (\phi_1, \cdots, \phi_N) = \exp \Bigl[ \sum_j \lambda(\varepsilon j ) \phi_j \Bigr] d\mu_0 \] 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 \eqref{he} 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{align*} \partial_{t}S(f_{t}) & = - D( \sqrt {f_{t}}), \\ S(f) & = \int f \log f \,d \mu_{0}. \end{align*} 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{equation}\label{db} \int_0^t D( \sqrt {f_{s}}) \,ds \le CN \end{equation} This in­form­a­tion alone is suf­fi­cient to es­tab­lish that the solu­tion of \eqref{1} 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 \eqref{dyn} so that the gen­er­at­or \( L \) be­comes the sym­met­ric op­er­at­or with Di­rich­let form \begin{align} \label{non} D(f) : & = -\int f \,L f \, d \mu_{0} \\ & = \sum_{ j=1}^N \int a (\phi_{j}, \phi_{j+1}) \Bigl( \frac {\partial f} { \partial \phi_{j}} - \frac {\partial f} { \partial \phi_{j+1}} \Bigr)^{2} d \mu_{0}. \nonumber \end{align} The cur­rent can be com­puted eas­ily, and is giv­en (up to scale factors) by \begin{equation} 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{equation} To es­tab­lish the hy­dro­dynam­ic­al lim­it, it is now re­quired to prove that, when \( x= \varepsilon j \), in some sense \( w_j \) can be re­placed by \begin{equation}\label{fl0} w_{j} = D(\rho(x, t)) [\phi_{j+1} - \phi_{j} ] \end{equation} 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{equation}\label{fd} w_{j} = D(\rho(x, t)) [\phi_{j+1} - \phi_{j} ] + L F \end{equation} where \( F \) is some loc­al func­tion of \( \phi_{j} \) and \( L \) is the gen­er­at­or giv­en by \eqref{non}. 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 \eqref{fd} 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 \eqref{fd} 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 \eqref{db}, one only has to solve \eqref{fd} 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{equation} \int w_{0}\, d \mu_{\lambda} = 0 \end{equation} 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 \eqref{fd}. 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 \eqref{1} 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 \eqref{1} 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 \eqref{fd}.


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[9]S. R. S. Varadhan: “Scal­ing lim­its for in­ter­act­ing dif­fu­sions,” Comm. Math. Phys. 135 : 2 (1991), pp. 313–​353. MR 1087387 Zbl 0725.​60085 article

[10]S. R. S. Varadhan: “En­tropy meth­ods in hy­dro­dynam­ic­al scal­ing,” pp. 103–​112 in Math­em­at­ic­al phys­ics X (Leipzig, 30 Ju­ly–9 Au­gust, 1991). Edi­ted by K. Schmüdgen. Spring­er (Ber­lin), 1992. MR 1386400 Zbl 0947.​82510 incollection

[11]S. R. S. Varadhan: “En­tropy meth­ods in hy­dro­dynam­ic scal­ing,” pp. 112–​145 in Nonequi­lib­ri­um prob­lems in many-particle sys­tems (Montec­at­ini, Italy, June 15–27, 1992). Edi­ted by L. Ark­eryd. Lec­ture Notes in Math­em­at­ics 1551. Spring­er (Ber­lin), 1993. MR 1296260 Zbl 0791.​60098 incollection

[12]S. R. S. Varadhan: “Re­l­at­ive en­tropy and hy­dro­dynam­ic lim­its,” pp. 329–​336 in Stochast­ic pro­cesses. Edi­ted by S. Cam­banis. Spring­er (New York), 1993. MR 1427330 Zbl 0790.​60094 incollection

[13]S. Olla, S. R. S. Varadhan, and H.-T. Yau: “Hy­dro­dynam­ic­al lim­it for a Hamilto­ni­an sys­tem with weak noise,” Comm. Math. Phys. 155 : 3 (1993), pp. 523–​560. MR 1231642 Zbl 0781.​60101 article

[14]S. R. S. Varadhan: “Non­lin­ear dif­fu­sion lim­it for a sys­tem with nearest neigh­bor in­ter­ac­tions. II,” pp. 75–​128 in Asymp­tot­ic prob­lems in prob­ab­il­ity the­ory: stochast­ic mod­els and dif­fu­sions on fractals (Sanda/Kyoto, 1990), vol. 1. Edi­ted by K. D. El­worthy and N. Ike­da. Pit­man Re­search Notes in Math­em­at­ics 283. Long­man Sci­entif­ic & Tech­nic­al (Har­low), 1993. MR 1354152 Zbl 0793.​60105 incollection

[15]S. R. S. Varadhan: “Reg­u­lar­ity of self-dif­fu­sion coef­fi­cient,” pp. 387–​397 in The Dynkin Fest­s­chrift: Markov pro­cesses and their ap­plic­a­tions. Edi­ted by M. I. Freĭdlin. Pro­gress in Prob­ab­il­ity 34. Birkhäuser (Bo­ston, MA), 1994. MR 1311731 Zbl 0822.​60089 incollection

[16]S. R. S. Varadhan: “En­tropy meth­ods in hy­dro­dynam­ic scal­ing,” pp. 196–​208 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Zürich, Au­gust 3–11, 1994), vol. I. Edi­ted by S. D. Chat­terji. Birkhäuser (Basel), 1995. MR 1403922 Zbl 0872.​60081 inproceedings

[17]S. R. S. Varadhan: “Self-dif­fu­sion of a tagged particle in equi­lib­ri­um for asym­met­ric mean zero ran­dom walk with simple ex­clu­sion,” Ann. Inst. H. Poin­caré Probab. Stat­ist. 31 : 1 (1995), pp. 273–​285. MR 1340041 Zbl 0816.​60093 article

[18]S. R. S. Varadhan: “The com­plex story of simple ex­clu­sion,” pp. 385–​400 in Itô’s stochast­ic cal­cu­lus and prob­ab­il­ity the­ory. Edi­ted by N. Ike­da, S. Watanabe, M. Fukushi­ma, and H. Kunita. Spring­er (Tokyo), 1996. MR 1439538 Zbl 0866.​60092 incollection

[19]C. Land­im, S. Se­th­ura­man, and S. Varadhan: “Spec­tral gap for zero-range dy­nam­ics,” Ann. Probab. 24 : 4 (1996), pp. 1871–​1902. MR 1415232 Zbl 0870.​60095 article

[20]S. R. S. Varadhan: “Non­gradi­ent mod­els in hy­dro­dynam­ic scal­ing,” pp. 397–​416 in Ana­lys­is, geo­metry and prob­ab­il­ity. Edi­ted by R. Bha­tia. Texts and Read­ings in Math­em­at­ics 10. Hindus­tan Book Agency (Del­hi), 1996. MR 1477705 Zbl 0977.​60099 incollection

[21]S. R. S. Varadhan and H. T. Yau: “Scal­ing lim­its for lat­tice gas mod­els,” pp. 39–​43 in Stochast­ic pro­cesses and func­tion­al ana­lys­is (River­side, CA, Novem­ber 18–20, 1994). Edi­ted by J. A. Gold­stein, N. E. Gret­sky, and J. J. Uhl. Lec­ture Notes in Pure and Ap­plied Math­em­at­ics 186. Dek­ker (New York), 1997. MR 1440413 Zbl 0884.​60090 incollection

[22]S. R. S. Varadhan and H.-T. Yau: “Dif­fus­ive lim­it of lat­tice gas with mix­ing con­di­tions,” Asi­an J. Math. 1 : 4 (December 1997), pp. 623–​678. MR 1621569 Zbl 0947.​60089 article

[23]J. Quastel, F. Reza­khan­lou, and S. R. S. Varadhan: “Large de­vi­ations for the sym­met­ric simple ex­clu­sion pro­cess in di­men­sions \( d\geq 3 \),” Probab. The­ory Re­lated Fields 113 : 1 (1999), pp. 1–​84. MR 1670733 Zbl 0928.​60087 article

[24]S. R. S. Varadhan: “Large de­vi­ations for in­ter­act­ing particle sys­tems,” pp. 373–​383 in Per­plex­ing prob­lems in prob­ab­il­ity. Edi­ted by M. Bramson and R. Dur­rett. Pro­gress in Prob­ab­il­ity 44. Birkhäuser (Bo­ston, MA), 1999. MR 1703141 Zbl 0941.​60096 incollection

[25]S. R. S. Varadhan: “In­fin­ite particle sys­tems and their scal­ing lim­its,” pp. 306–​315 in Math­em­at­ic­al phys­ics 2000. Edi­ted by A. Fo­kas, A. Grigory­an, T. Kibble, and B. Zegar­l­in­ski. Im­per­i­al Col­lege Press (Lon­don), 2000. MR 1773051 Zbl 1028.​82516 incollection

[26]S. Se­th­ura­man, S. R. S. Varadhan, and H.-T. Yau: “Dif­fus­ive lim­it of a tagged particle in asym­met­ric simple ex­clu­sion pro­cesses,” Comm. Pure Ap­pl. Math. 53 : 8 (August 2000), pp. 972–​1006. MR 1755948 Zbl 1029.​60084 article

[27]S. R. S. Varadhan: “Lec­tures on hy­dro­dynam­ic scal­ing,” pp. 3–​40 in Hy­dro­dynam­ic lim­its and re­lated top­ics (Toronto, Oc­to­ber 7–10, 1998). Edi­ted by S. Feng, A. T. Lawn­iczak, and S. R. S. Varadhan. Fields In­sti­tute Com­mu­nic­a­tions 27. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2000. MR 1798641 Zbl 1060.​82514 incollection

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[29]C. Land­im, S. Olla, and S. R. S. Varadhan: “Asymp­tot­ic be­ha­vi­or of a tagged particle in simple ex­clu­sion pro­cesses,” Bol. Soc. Brasil. Mat. (N.S.) 31 : 3 (2000), pp. 241–​275. MR 1817088 Zbl 0983.​60100 article

[30]S. R. S. Varadhan: “Scal­ing lim­its of large in­ter­act­ing sys­tems,” pp. 144–​158 in The math­em­at­ic­al sci­ences after the year 2000 (Beirut, 11–15 Janu­ary, 1999). Edi­ted by K. Bit­ar, A. Chamsed­dine, and W. Sabra. World Sci­entif­ic (River Edge, NJ), 2000. MR 1799447 Zbl 1112.​82320 incollection

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[32]S. R. S. Varadhan: “Large de­vi­ation and hy­dro­dynam­ic scal­ing,” pp. 265–​286 in Tanigu­chi con­fer­ence on math­em­at­ics Nara ’98 (Nara, Ja­pan, Decem­ber 15–20, 1998). Edi­ted by M. Maruyama and T. Sunada. Ad­vanced Stud­ies in Pure Math­em­at­ics 31. Math­em­at­ic­al So­ci­ety of Ja­pan (Tokyo), 2001. MR 1865096 Zbl 1006.​60019 incollection

[33]C. Land­im, S. Olla, and S. R. S. Varadhan: “Fi­nite-di­men­sion­al ap­prox­im­a­tion of the self-dif­fu­sion coef­fi­cient for the ex­clu­sion pro­cess,” Ann. Probab. 30 : 2 (2002), pp. 483–​508. MR 1905849 Zbl 1018.​60097 article

[34]S. R. S. Varadhan: “Particle sys­tems and par­tial dif­fer­en­tial equa­tions,” pp. 217–​222 in Lec­tures on par­tial dif­fer­en­tial equa­tions: Pro­ceed­ings in hon­or of Louis Niren­berg’s 75th birth­day. Edi­ted by S.-Y. A. Chang, C.-S. Lin, and H.-T. Yau. New Stud­ies in Ad­vanced Math­em­at­ics 2. In­ter­na­tion­al Press (Somerville, MA), 2003. MR 2055850 Zbl 1167.​82351 incollection

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[36]C. Land­im, S. Olla, and S. R. S. Varadhan: “On vis­cos­ity and fluc­tu­ation-dis­sip­a­tion in ex­clu­sion pro­cesses,” J. Stat­ist. Phys. 115 : 1–​2 (2004), pp. 323–​363. MR 2070098 Zbl 1157.​82355 article

[37]C. Land­im, S. Olla, and S. R. S. Varadhan: “Dif­fus­ive be­ha­viour of the equi­lib­ri­um fluc­tu­ations in the asym­met­ric ex­clu­sion pro­cesses,” pp. 307–​324 in Stochast­ic ana­lys­is on large scale in­ter­act­ing sys­tems. Edi­ted by T. Fun­aki and H. Os­ada. Ad­vanced Stud­ies in Pure Math­em­at­ics 39. Math­em­nat­ic­al So­ci­ety of Ja­pan (Tokyo), 2004. MR 2073338 Zbl 1080.​60094 incollection

[38]S. R. S. Varadhan: “Large de­vi­ations and scal­ing lim­it,” Lett. Math. Phys. 88 : 1–​3 (2009), pp. 175–​185. MR 2512145 Zbl 1185.​60024 article

[39]S. R. S. Varadhan: “Scal­ing lim­its,” pp. 247–​262 in Per­spect­ives in math­em­at­ic­al sci­ences, vol. I: Prob­ab­il­ity and stat­ist­ics. Edi­ted by N. S. N. Sastry, T. S. S. R. K. Rao, M. Delampady, and B. Ra­jeev. Stat­ist­ic­al Sci­ence and In­ter­dis­cip­lin­ary Re­search 7. World Sci­entif­ic (Hack­en­sack, NJ), 2009. MR 2581747 incollection

[40]S. Se­th­ura­man and S. R. S. Varadhan: Large de­vi­ations for the cur­rent and tagged particle in 1D nearest-neigh­bor sym­met­ric simple ex­clu­sion. Pre­print, January 2011. ArXiv 1101.​1479 techreport