Extreme current fluctuations of boundary-driven systems in the large- limit
Current fluctuations in boundary-driven diffusive systems are, in many cases, studied using hydrodynamic theories. Their predictions are then expected to be valid for currents which scale inversely with the system size. To study this question in detail, we introduce a class of large- models of one-dimensional boundary-driven diffusive systems, whose current large deviation functions are exactly derivable for any finite number of sites. Surprisingly, we find that for some systems the predictions of the hydrodynamic theory may hold well beyond their naive regime of validity. Specifically, we show that, while a symmetric partial exclusion process exhibits non-hydrodynamic behaviors sufficiently far beyond the naive hydrodynamic regime, a symmetric inclusion process is well described by the hydrodynamic theory for arbitrarily large currents. We conjecture, and verify for zero-range processes, that the hydrodynamic theory captures the statistics of arbitrarily large currents for all models where the mobility coefficient as a function of density is unbounded from above. In addition, for the large- models, we prove the additivity principle under the assumption that the large deviation function has no discontinuous transitions.
- I Introduction
- II Large- models
- III Current large deviations in the SPEP
- IV Current large deviations of SIP
- V Criterion for persistent hydrodynamic behaviors
- VI Conclusions
- A Current fluctuations in the hydrodynamic limit
- B Path Integral representation of the CGF
- C Finite- corrections to the CGF for the SPEP arising from space-time fluctuations
- D Symmetric zero-range process
One of the most fundamental ways to characterize the steady state of a system is through the statistical properties of currents. These have been studied both in and out of equilibrium and in both classical Derrida (2007) and quantum systems Pilgram et al. (2003); Esposito et al. (2009); Genway et al. (2014). Recently, much progress has been achieved in understanding the statistics of time-averaged currents, which are encoded in a corresponding large deviation functions (LDF), of boundary-driven diffusive systems in one dimension Derrida et al. (2004); Bodineau and Derrida (2004); Bertini et al. (2005a, 2006); Harris et al. (2005); Imparato et al. (2009); Lecomte et al. (2010); Shpielberg and Akkermans (2015) as well as in other geometries Bodineau et al. (2008); Akkermans et al. (2013). Whereas exact microscopic solutions are often available for bulk-driven systems Derrida and Lebowitz (1998); Prolhac and Mallick (2009); Lazarescu and Mallick (2011); Mallick (2011); Lazarescu (2013, 2015); Ayyer (2015), the results for boundary-driven systems largely rest on the application of a hydrodynamic approach termed the macroscopic fluctuation theory (MFT) Spohn (1991); Bertini et al. (2002); Jordan et al. (2004); Bertini et al. (2015), with the notable exception of Lazarescu (2013).
Being a hydrodynamic theory, the MFT is naively expected to yield the correct statistics of currents only when the current fluctuations are small enough for the hydrodynamic description to be valid. For example, consider a single-species diffusive system on the line , where denotes the length of the system. After coarse-graining and a diffusive rescaling ( and 111These notations indicate that the rescaled variables are defined as and , and then renamed as and , respectively. Other notations for rescaling schemes should be interpreted similarly.), the hydrodynamic equation takes the form
with the coarse-grained density and the coarse-grained current. Since we are interested in the limit, this equation is not well defined for which before the rescaling is not of the order of . Thus, the statistics of currents obtained by the MFT are reliable only for current fluctuations of the order of . The same conclusion can be reached by another argument more directly based on the MFT, which is discussed in Appendix A.
In this paper we study the validity of the hydrodynamic approach in regions where it is expected to fail. Quite surprisingly, we find that there are classes of models where the hydrodynamic approach captures the statistics of currents much beyond its naive regime of validity. We give a simple explanation for this phenomena and based on it argue that this behavior is expected to be generic when the mobility diverges with the density of particles.
To obtain these results, we study current LDFs of boundary-driven systems whose lattice structure is preserved, keeping a finite number of sites . Since the exact current LDFs of microscopic lattice models are difficult to obtain (with the exception of the zero-range-process Harris et al. (2005)), we consider a little-studied class of coarse-grained models, which we term large- models. A large- model consists of a one-dimensional chain of boxes, each of which holds a macroscopically large number of particles (controlled by ) and which relaxes instantaneously to local equilibrium. As such, it retains the lattice structure even after coarse-graining and can be thought of as an analog of the “boxed models” studied in Bunin et al. (2013); Kafri (2015). In a manner similar to models of population dynamics Elgart and Kamenev (2004); Meerson and Sasorov (2011) and lattice spin models in the large-spin limit Tailleur et al. (2007, 2008), we rescale dynamical variables and hopping rates of the model by powers of . This allows us to apply the standard saddle-point techniques in the limit.
Thanks to simplifications arising from the assumption of a macroscopic number of particles at each box (site), the current LDFs of our large- models are exactly derivable even for a finite system with any number of sites . By comparing the tail behaviors of the current LDFs in the large- limit with the predictions of the MFT approach, we can observe how and when non-hydrodynamic behaviors start to emerge. Interestingly, our formulation also shows that the same microscopic dynamics may produce different macroscopic models depending on how the microscopic variables are scaled with .
We note that there were previous studies on models with multiple particles per site, such as partial exclusion processes Schütz and Sandow (1994), inclusion processes Giardinà et al. (2007), or both Giardinà et al. (2010); Carinci et al. (2013). These studies obtained exact expressions for particle density correlations on a finite lattice with sites. The corresponding density large deviations were studied in Tailleur et al. (2007, 2008), but only after a gradient expansion in the limit that washes away the lattice structure. To our knowledge, large deviation properties of these models at finite have not been properly explored 222We note that there was a previous attempt to calculate the current LDF of a discrete system by applying a saddle-point approximation directly to the microscopic model Imparato et al. (2009). This approximation, however, is not well controlled..
This paper is organized as follows. In Sec. II, we introduce two classes of large- models, which are the symmetric partial exclusion process (SPEP) and the symmetric inclusion process (SIP). It is shown that the latter becomes equivalent to the well-studied Kipnis–Marchioro–Presutti (KMP) model Kipnis et al. (1982); Bertini et al. (2005b) after an appropriate rescaling by . In Sec. III, we study current large deviations of the SPEP, which exhibits non-hydrodynamic behaviors for current fluctuations sufficiently far beyond the naive hydrodynamic regime expected by the argument given above. In addition, we also discuss the validity of the additivity principle. In Sec. IV, we analyze current large deviations of the SIP for different large- limits, which in all cases exhibit hydrodynamic behaviors for arbitrarily large current fluctuations. Based on these results, in Sec. V we propose a criterion for the persistence of hydrodynamic current fluctuations in the non-hydrodynamic regime, and confirm its validity for the symmetric zero-range process. Finally, we summarize our results and conclude in Sec. VI.
Ii Large- models
We now turn to introduce the large- versions of the SPEP and the SIP. Starting with the SPEP the microscopic model is defined and used to obtain a path-integral representation for the current cumulant generating function (CGF) along with the prescription for calculating it in the large- limit. The hydrodynamic limit of the model is then presented for completeness. The section closes by giving the corresponding results for the class of SIP models.
ii.1 Microscopic dynamics
The models are defined on a one-dimensional chain of boxes which are in contact with two particle reservoirs denoted by and (see Fig. 1 for an illustration). Each box is assumed to be in local equilibrium so that the state of box is completely specified by the number of particles , for . A particle hops from a box to an adjacent one with a rate (in arbitrary units) given by
which reflects exclusion (‘attractive’) interactions between particles in the SPEP (SIP). It is clear that for the SPEP the range of is bounded from above and below (), while for the SIP is only bounded from below (). The hopping rates at the boundaries are defined similarly as:
If the system is coupled only to reservoir (reservoir ), the average number of particles in each box relaxes to () as determined by and ( and ). In what follows, we fix the contact rates to the reservoirs through , for the SPEP, and , for the SIP. The parameters and thus fully describe the coupling with the reservoirs:
This choice provides simpler expressions in the results presented below, without affecting the large- hydrodynamic behavior.
With these definitions it is natural to introduce density variables according to
and rescale time as . Then the evolution of the average density profile, taken over some initial distribution and denoted by angular brackets, satisfies
for any with and . We note that the discrete diffusion equation (6) is also known to hold exactly for the standard Symmetric Simple Exclusion Process (SSEP), which corresponds to the SPEP with .
Under this rescaling, for the SPEP, is naturally interpreted as the capacity of each box. On the other hand, for the SIP the number of particles is not bounded from above. Therefore, does not admit a natural interpretation without specifying how both and scale with . In fact, one can choose an alternate scaling and define densities for the SIP as
with rescaled by as above and (the rationale behind this constraint will become clear below). It is straightforward to check that (6) is then unchanged. Interestingly, these two scaling choices for the SIP, as we show below, lead to different macroscopic theories. In what follows, when we also study the SIP rescaled by (7) and refer to it as SIP(1+), in contrast to the SIP(1) whose scaling is defined in (5).
ii.2 SPEP – current CGF and hydrodynamic limit
Our interest is in calculating the current CGF which encodes the statistics of the time-averaged density current . We can obtain , for example, by measuring the flux of particles from box to reservoir during an interval . The CGF is then defined through
where the average, denoted by angular brackets, is taken with fixed and , and is conjugate to the current . Using standard methods (see Appendix B), we can write a path-integral representation of the CGF
with the density vector and the auxiliary ‘momentum’ vector. For the SPEP, the Hamiltonian is given by
When is very large (in the sense of ), the large- CGF can be obtained using saddle-point asymptotics
with the infimum taken over trajectories of and . As advertised above, this approximation requires only to be a large parameter, so its predictions hold for any value of . The minimization principle (11) is similar to that of the MFT approach Bertini et al. (2015) for the SSEP, with , instead of , playing the role of the large parameter governing the saddle-point. This allows us to keep track of the lattice structure at any finite .
Assuming that the minimizing trajectory is time-independent, the saddle-point equations are given by
The solutions of these equations, which we denote by and , are typically called the optimal profiles which support the current fluctuation . Then the current CGF is obtained from (11) as
The additivity principle, proposed in Bodineau and Derrida (2004) (also independently studied in Jordan et al. (2004)), implies that the above assumption is applicable for any value of . Although counterexamples were found in periodic bulk-driven systems Bertini et al. (2005a); Bodineau and Derrida (2005); Bertini et al. (2006); Hurtado and Garrido (2011); Espigares et al. (2013), the principle was analytically shown to be true for any open boundary-driven diffusive system with a constant diffusion coefficient and a quadratic mobility coefficient Imparato et al. (2009) — without ruling out possible discontinuous transitions, which in turn were numerically discarded in Hurtado and Garrido (2009) for a specific model related to the SIP. As shown below, both the SPEP and the SIP correspond to this class of systems in the hydrodynamic limit. Thus we expect that the same principle is also applicable to our large- models, and discuss arguments supporting its validity in Sec. III.3 and Appendix C.
Finally, we show that under appropriate assumptions our large- models are well described by hydrodynamic theories. To see this, we first apply a diffusive scaling in terms of , which involves writing the position of box as (with the lattice spacing set to one) and rescaling time by . We also assume that differences between adjacent boxes, namely and , scale as . Then, in the limit, the gradients and are well defined, and (9) can be approximated as
Here the Hamiltonian , which is no longer dependent on , is now a functional of continuous profiles and . The functional typically has the form of
with the diffusion coefficient and the mobility coefficient. For the SPEP, these coefficients are given by
respectively. We note that this is bounded from above, with the maximum value given by . Meanwhile, the rescaling of time speeds up the microscopic dynamics, so the leftmost () and rightmost () boxes equilibrate with the coupled reservoirs (see e.g. Appendix B.2 of Ref. Tailleur et al. (2008)). Hence, the spatial boundary conditions are given by
whose dependence on keeps a function of .
In what follows we list the corresponding sets of results for the SIP(1) and the SIP(1+).
ii.3 SIP(1) – current CGF and hydrodynamic limit
ii.4 Sip(1+) – current CGF and hydrodynamic limit
For the SIP(1+) we similarly find, using the notation of (9),
The hydrodynamic description of this model in the large- limit is given by (15) with
where is again not bounded from above. These transport coefficients are also shared by the Kipnis–Marchioro–Presutti (KMP) model of heat conduction Kipnis et al. (1982); Bertini et al. (2005b). It is notable that the same microscopic model produces different macroscopic behaviors depending on the reservoir properties.
Iii Current large deviations in the SPEP
In what follows we first show that the scaled CGF of the time-averaged current in the SPEP in the large- limit is given by
Note that although the result depends explicitly on the sign of , it is straightforward to verify that it is an analytic function of . After deriving this result, we compare (22) to the predictions of the hydrodynamic theory. As we show, for large enough currents the two theories, as one might expect using the simple argument of the introduction, do not agree. Finally, we discuss finite- effects and their implications on the additivity principle.
iii.1 Derivation of the scaled CGF
As stated above, assuming additivity, the problem of calculating the CGF in the large- limit is reduced to solving (12). To do this it is useful to use the canonical transformation Derrida et al. (2002); Tailleur et al. (2007, 2008)
which can also be written as
Then the Hamiltonian in the new set of coordinates, , is given by
where and .
Note that the canonical transformation also adds temporal boundary conditions to the action which can be ignored in the limit. The scaled CGF is then given by:
where are solutions of
In what follows, we solve these equations using the methods used in Imparato et al. (2009). To avoid cumbersome expressions we drop the notation from the optimal profiles and use . First, we choose the Ansatz
where and are undetermined constants. It is easy to check that this Ansatz satisfies (28) for . Then the constants and are determined by the remaining saddle-point equations
These equations imply
from which we obtain
Then one can show that has the form of
Given , (33) is solved by
Thus we have found and up to the undetermined signs of and . These signs can be fixed by noting that the optimal density profile must always be nonnegative and that the CGF must vanish at . Without loss of generality, for the optimal profiles are given by
We note that and are negative for the intermediate range and nonnegative otherwise. The results for are easily obtained by a sign change and an exchange of and . Using these results with (27), after some algebra one obtains (22).
iii.2 Comparison with hydrodynamic results
We now compare the results of the large- limit with the predictions of the hydrodynamic theory. The latter has been derived in Bodineau and Derrida (2004); Imparato et al. (2009) (for the SSEP which shares the same hydrodynamic theory) and can also be obtained by holding fixed in (22) and taking the large limit. The expression is given by
and the convergence to it is illustrated in Fig. 2. In fact, one can show analytically that
The sign of the leading correction term indicates that the lattice structure increases the magnitude of the current fluctuations.
To check the validity of the hydrodynamic predictions we next increase as . This gives
This indicates that, as one would naively expect, the hydrodynamic description fails for sufficiently large currents. The threshold separating the hydrodynamic regime from the non-hydrodynamic regime is given by (see Fig. 3).
As we later show, there are other models where the predictions of the hydrodynamic theory hold well beyond the naive expectation. To this end it is useful to see in detail how the predictions of the hydrodynamic limit fail for the SPEP. To do this, we note that Hamilton’s equation takes the form of
where is the current from box to box . The time-averaged current can be expressed in terms of the optimal profiles (again we drop the notation) as
Since the mean value always scales as , large values of are always dominated by the fluctuation (see Meerson and Sasorov (2014) for a similar observation). Next, note that, as shown in Fig. 4, a large is supported by a plateau of the density profile close to and a slope of the momentum profile which grows with (and hence with ). In addition, as indicated by the data collapses in Fig. 5, the momentum profile has the scaling form
This implies that
If , the momentum gradient decreases with . Then we can approximate as
whose integral form suggests that the current is blind to the lattice structure for any . In other words, the current does not feel any difference between the case (which can be considered as proper hydrodynamic regime) and the case . Thus its fluctuations show hydrodynamic behaviors in both cases. On the other hand, if , the momentum gradient increases with . Then the approximate (44) becomes invalid, and the current becomes sensitive to the lattice structure. Thus , which corresponds to by (III.2), is the threshold separating the hydrodynamic regime from the non-hydrodynamic one. We note that this threshold is larger than what one would naively expect from the simple argument given in Sec. I, i.e., .
iii.3 Finite- corrections and the validity of the additivity principle
In what follows, we analyze the leading finite- correction to the scaled CGF . This provides a useful tool for numerical corroboration of our analytical results, and confirms the stability of the time-independent saddle-point profiles. The latter thus supports the validity of the additivity principle for the SPEP.
As explained in Appendix C, one can integrate spatio-temporal fluctuations around the saddle-point optimal solutions. This is done by using a mapping (generalizing that of Ref. Lecomte et al. (2010)) between the CGF of the system with reservoirs at generic densities , and the CGF for reservoirs at densities . The resulting expression is finite and analytic, which proves that the additivity hypothesis is correct with respect to continuous phase transitions towards time-dependent profiles (which, if they had existed, would have implied an instability of , reflected in a singularity of the correction). The saddle-point contribution to the CGF is complemented by a correction:
with, denoting ,
where we defined .
We numerically confirm our theoretical predictions by implementing a finite- propagator of the SPEP conditioned on a given value of . The eigenvalue with the largest real part corresponds to the scaled CGF . As shown in Fig. 6, our theory correctly predicts the leading-order behaviors of .
We now detail how the large- limit (at fixed ) of (46) matches the MFT result obtained for the SSEP Appert-Rolland et al. (2008). The limit behavior of (46) is not immediately extractable; following a procedure described in Appendix C, one obtains
Here, with , we recognize the universal scaling function
as the one also arising in MFT Appert-Rolland et al. (2008) and Bethe-Ansatz Appert-Rolland et al. (2008); Prolhac and Mallick (2009) studies of current fluctuations. The large- limit (at fixed ) thus yields the same correction as in the MFT approach Imparato et al. (2009) for the SSEP. The universal scaling function is singular at a positive value of its argument, but this value is never reached for any real-valued in (47). This confirms, as in the MFT context, that the additivity principle holds at large .
Iv Current large deviations of SIP
In this section, we derive the scaled CGF of the time-averaged current of the SIP in the large- limit. It is given by
with the differences between and encoded in
Again, one can easily verify that does not have any singularity at . A comparison of this result with the hydrodynamic theory shows that for both and arbitrarily large current fluctuations are still correctly captured by the hydrodynamic theory, in contrast to the SPEP. We close the section with a discussion of finite- effects.
iv.1 Derivation of the scaled CGF
Similarly to the SPEP, we first transform into a more convenient form. This is done by the canonical transformation
which can also be written as
After this transformation, the Hamiltonian of the new variables is given by