Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

A gradient system is a triple \(({\mathbf{Q}},{{\mathcal {E}}},{{\mathcal {R}}})\) of a state space \({\mathbf{Q}}\), a functional \({{\mathcal {E}}}\) on \({\mathbf{Q}}\), and a dissipation potential \({{\mathcal {R}}}\). This triple defines in a unique way a differential equation for the evolution \(t\mapsto q(t)\) of the states, the so-called gradient-flow equation:which can be seen as a balance of thermodynamical forces, namely the potential restoring force \(-{\mathrm D}{{\mathcal {E}}}(q)\) and the viscous force \( \xi = {\mathrm D}_{\dot{q}} {{\mathcal {R}}}(q, \dot{q})\) induced by the rate \(\dot{q}\). Indeed, any functional dependence \(\xi = K(q,\dot{q})\) or \(\dot{q}=G(q,\xi )\) between the rate \(\dot{q}\) and the dual (viscous) friction force \(\xi \) is often called a kinetic relation. Gradient-flow equations are distinguished by two facts: These two conditions allow for a variational characterization for the gradient-flow equation (1.1), the so-called energy-dissipation principle, which is the basis of this work; see Sect. 2 for this and a more detailed description to gradient systems.

Using the Fenchel-Legendre transform, one can define a dual dissipation potential \({{\mathcal {R}}}^*(q,\xi )\) such that the kinetic relation can be written through any of the three equivalent conditionsWhile, for a given gradient system \(({\mathbf{Q}},{{\mathcal {E}}},{{\mathcal {R}}})\), the gradient-flow equation (1.1) is uniquely given and may be rewritten in the formthe opposite direction, however, shows a strong non-uniqueness for a given vector field \({\mathbf{V}}\) and a given energy \({{\mathcal {E}}}\); there may be many kinetic relations \({{{G}}}\) and even many dual dissipation potentials \({{\mathcal {R}}}^*\), such that \({\mathbf{V}}\) is generated as in (1.3).

We say that that the differential equation \(\dot{q} ={\mathbf{V}}(q)\) has the gradient structure \(({\mathbf{Q}},{{\mathcal {E}}},{{\mathcal {R}}})\) if \({\mathbf{V}}(q)={\mathrm D}_\xi {{\mathcal {R}}}^*(q, {-}{\mathrm D}{{\mathcal {E}}}(q))\). Adding such a gradient structure to a differential equation means to identify additional thermodynamical information that is no longer visible in the induced gradient-flow equation \(\dot{q}={\mathbf{V}}(q)\).

We first illustrate the concept of a gradient system with an example, in which a spring relaxes by moving a damper (a shock absorber), see Fig. 1. The state of the system is the spring displacement \(q\in {{\mathbb {R}}}\), the energy contained in the spring is \({{\mathcal {E}}}_1(q) :=kq^2/2\), and the spring exerts a force \(\xi \) equal to the negative derivative \(-{\mathrm D}{{\mathcal {E}}}_1(q) = -kq\) of the energy. The damper is defined by the property that its rate v of displacement is related to the force \(\xi \) on the damper by \(\mu v = \xi \), for some coefficient \(\mu >0\). By combining these two relations, we find the evolution equation for the state q,We identify Eq. (1.4) as the gradient flow-equation for \(({{\mathbb {R}}},{{\mathcal {E}}}_1,{{\mathcal {R}}}_1)\), when we observe that the damper relation \(\mu v = \xi \) can also be written in terms of a dissipation potential \({{\mathcal {R}}}_1(v) :=\mu v^2/2\) and its Legendre dual \({{\mathcal {R}}}_1^*(\xi ) :=\xi ^2/(2\mu )\). The dissipation potential \({{\mathcal {R}}}_1\) defines the kinetic relation \(\mu v = \xi \).

In this example, one readily recognizes a ‘classical’ spring energy in \({{\mathcal {E}}}_1(q) = kq^2/2\), and the quadratic form of \({{\mathcal {R}}}_1(v) = \mu v^2/2\) is a natural choice for a damper (see, e.g. [33, Ch. 5]). However, other gradient-flow formulations for the same evolution equation (1.4) exist, if \({{\mathcal {R}}}={{\mathcal {R}}}(q,v)\) may depend not only on the rate \(v =\dot{q}\) but also on the state q:All the systems \(({{\mathbb {R}}},{{\mathcal {E}}}_i,{{\mathcal {R}}}_i)\) generate the same equation (1.4) via \({\mathrm D}_v{{\mathcal {R}}}_i(q,\dot{q}) = -{\mathrm D}{{\mathcal {E}}}_i(q)\).

In fact, even in this simple scalar example, one can generate a wide variety of gradient systems for the same equation (1.4): take any smooth and convex \(\psi :{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) with \(\min \psi = \psi (0) = 0\), define \(\varphi (q) = -kq/\psi '(-kq/\mu )\) and \({{\mathcal {R}}}_\psi (q,v) :=\varphi (q)\psi (v)\), and then, the gradient system \(({{\mathbb {R}}},{{\mathcal {E}}}_1, {{\mathcal {R}}}_\psi )\) will generate Eq. (1.4). The two examples \({{\mathcal {R}}}_2\) and \({{\mathcal {R}}}_3\) above are both of this type.

These dissipation potentials might well be considered less ‘natural’ than \({{\mathcal {R}}}_1\). To start with, it is not obvious which modeling arguments would lead to the kinetic relations of \({{\mathcal {R}}}_2\) and \({{\mathcal {R}}}_3\), which areIn addition, a definition like that of \({{\mathcal {R}}}_3\) is dimensionally inconsistent, since arguments of the exponential function should be dimensionless. Both these problems are related to a deeper and more troubling problem: the dissipation potentials depend not only on \(\mu \) but also on k, implying that the kinetic relation generated by \({{\mathcal {R}}}_2\) or \({{\mathcal {R}}}_3\), which is supposed to characterize the damper, depends on the strength k of the spring. This is an unsatisfactory situation: we consider the spring and the damper to be two independent objects, and their mathematical characterizations should therefore also be independent.

This example points toward the problem that we aim to solve in this paper. This problem arises especially when taking limits of gradient systems in some parameter \(\varepsilon \rightarrow 0\); in such limits, it is unavoidable that the limiting dissipation potential depends on the state q as well as the rate of change v. As a result, the limiting evolution equation will have many gradient-flow structures, as in the example above. It turns out that one of the most common concepts used to define limits of gradient systems, which we call ‘simple EDP convergence’ in this paper and which we explain below, often selects limit dissipation potentials that are ‘unhealthy’ in the same way as \({{\mathcal {R}}}_2\) and \({{\mathcal {R}}}_3\) are ‘unhealthy’: they depend on aspects of the energy in an unsatisfactory way.

The aim of this paper is to construct alternative convergence concepts that lead to limiting gradient systems that are more ‘natural’ or ‘healthy.’ What we mean by these terms will become clear below, but first we consider an example to further illustrate the problem.

In Sect. 3, we study the following example in detail. Consider a family of gradient systems \(({{\mathbb {R}}},{{\mathcal {E}}},{{\mathcal {R}}}_\varepsilon )\), indexed by \(\varepsilon >0\), where \({{\mathcal {E}}}\) is some smooth \(\varepsilon \)-independent function, andHere \(\mu \in {\mathrm C}^0({{\mathbb {R}}}^2)\) is positive and 1-periodic in the second variable. For this ‘wiggly dissipation’ system, the gradient-flow equation takes the formAn example of a solution is given in Fig. 2.

We show in Sect. 3 that for \(\varepsilon \rightarrow 0\), the solutions \(q_\varepsilon \) of (1.5) converge to limit functions \(q_0\) that solve the limiting equationIn fact, \(({{\mathbb {R}}},{{\mathcal {E}}},{{\mathcal {R}}}_\varepsilon )\) converges in the simple EDP sense (defined in Sect. 2.3) to a limiting system \(({{\mathbb {R}}},{{\mathcal {E}}},\widetilde{{\mathcal {R}}}_0)\), where and \({{\mathcal {M}}}_0\) is defined via We verify explicitly in Sect. 3 that the system \(({{\mathbb {R}}},{{\mathcal {E}}}, \widetilde{{\mathcal {R}}}_0)\) indeed generates Eq. (1.6), i.e., thatHowever, this limiting dissipation potential \(\widetilde{{\mathcal {R}}}_0\) suffers from the same problem as \({{\mathcal {R}}}_2\) and \({{\mathcal {R}}}_3\) above: it depends explicitly on the energy function \({{\mathcal {E}}}\), as is clear from (1.7a). If one repeats the simple EDP convergence theorem for a perturbed energy \({{\mathcal {E}}}+{{\mathcal {F}}}\) with an arbitrary tilting function \({{\mathcal {F}}}\), then \({{\mathcal {F}}}\) propagates into the formula (1.7a) for \(\widetilde{{\mathcal {R}}}_0\); changing the energy thus leads to a different dissipation potential \(\widetilde{{\mathcal {R}}}_0\). As above, we consider this unsatisfactory, since the energy driving the system is conceptually separate from the mechanism for dissipating that energy.

In contrast, if we disregard the fact that Eq. (1.6) arises as a limit, and consider it as an isolated system, then we might conjecture a gradient structure with the effective dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}(q,v) :=\bar{\mu }(q)\, v^2/2\) instead. Indeed, combined with the energy \({{\mathcal {E}}}\) this potential \( {{\mathcal {R}}}_{\mathrm {eff}}\) also generates equation (1.6); it is much simpler to interpret than \(\widetilde{{\mathcal {R}}}_0\), and most importantly, it does not depend on \({{\mathcal {E}}}\).

These examples show that we have on one hand an unsatisfactory convergence result, in which \(({{\mathbb {R}}},{{\mathcal {E}}},\widetilde{{\mathcal {R}}}_0)\) is proved to arise as the unique limit of the family \(({{\mathbb {R}}},{{\mathcal {E}}},{{\mathcal {R}}}_\varepsilon )\) in the simple EDP sense, but this limit is unsatisfactory as a description of a gradient system.

On the other hand, the alternative dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}\) generates the same limit equation and does not suffer from the philosophical problems associated with \(\widetilde{{\mathcal {R}}}_0\). Its only drawback is that the system \(({{\mathbb {R}}}, {{\mathcal {E}}}, {{\mathcal {R}}}_{\mathrm {eff}})\) is not the limit of the family \(({{\mathbb {R}}},{{\mathcal {E}}},{{\mathcal {R}}}_\varepsilon )\) in the simple EDP sense.

As mentioned above, these observations strongly suggest seeking alternative convergence concepts for gradient systems, which should generate limiting potentials that do not depend on the limiting energy. Specifically, we will seek convergence concepts—let us indicate them with ‘\(\square \)’—that have the following property: ifthen for all \({{\mathcal {F}}}\in {\mathrm C}^1({\mathbf{Q}})\) we also havewhere the dissipation potential \({{\mathcal {R}}}_0\) in (1.9) is the same as in (1.8) and therefore does not depend on the tilt function \({{\mathcal {F}}}\).

Indeed, the two new concepts that we introduce in Sect. 2.6 both have this property, and we show in Sect. 3 that, by applying one of these convergence concepts, we indeed find the more natural dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}\) rather than \(\widetilde{{\mathcal {R}}}_0\).

Our aim of deriving ‘healthy’ limiting gradient systems could also be formulated as the challenge of deriving effective kinetic relations. We already introduced a kinetic relation as a relation between a force \(\xi \) and a rate \(v = \dot{q}\). An important class of such kinetic relations arises naturally in gradient systems, since dissipation potentials \({{\mathcal {R}}}\) define kinetic relations via the three equivalent relations (1.2).

In view of the Young-Fenchel inequality \( {{\mathcal {R}}}(q,v) + {{\mathcal {R}}}^*(q,\xi ) \ge \langle \xi , v \rangle \), which holds generally for Legendre conjugate pairs \(({{\mathcal {R}}},{{\mathcal {R}}}^*)\), and the third formulation in (1.2), we define the contact set as the set of pairs \((v,\xi )\):This set \({{\mathcal {C}}}\) characterizes the pairs of rates v and forces \(\xi \) that are admissible to the system and thus determine the kinetic relation. As was already mentioned, the equation generated by the gradient system can be viewed as the result of applying the kinetic relation \((v,\xi )\in {{\mathcal {C}}}_{{{\mathcal {R}}}\oplus {{\mathcal {R}}}^*}(q)\) to a context where the force \(\xi \) is generated by the potential \({{\mathcal {E}}}\):Kinetic relations appear throughout physics and mechanics. Well-known examples are Stokes’ law \(\xi =6\pi \eta R\, v\) for the drag force \(\xi \) on a sphere dragged through a viscous fluid (where \(\eta \) is the dynamic viscosity and R the radius of the sphere), power-law viscous relationships of the form \(\xi = c |v|^{p-1} v\), and Coulomb friction \(\xi \in c {\text {Sign}}(v)\), where \({\text {Sign}}\) is the subdifferential of the absolute-value function. These examples show that the relationship may be linear or nonlinear, and single- or multi-valued. A priori, there is no reason why a kinetic relation should be the graph of the derivative of a dissipation potential, but here we are interested in the ones that do have that property. The reasoning for the restriction of kinetic relations in form of subdifferentials, i.e., \(\xi ={\mathrm D}_v {{\mathcal {R}}}(q,v)\), is twofold. First, they define gradient systems and thus lead to variational characterizations for the gradient flow (see Sect. 2.2). Secondly, dissipation potentials arise naturally from thermodynamic principles derived from microscopic stochastic models via large deviation principles; see Sect. 6 and [2, 28, 29, 35]. Moreover, Onsager’s fundamental symmetry relation \({{\mathbb {G}}}={{\mathbb {G}}}^*\) for the linear kinetic relation \(\xi = {{\mathbb {G}}}v\) (see [30]) is equivalent to the existence of a (quadratic) dissipation potential \({{\mathcal {R}}}(v)=\frac{1}{2}\langle {{\mathbb {G}}}v,v\rangle \).

We now turn to the challenge of deriving effective kinetic relations. We are given a family of kinetic relations parametrized by \(\varepsilon \). The interpretation of \(\varepsilon \) as a small parameter, or a small scale, often implies that there are natural ‘macroscopic,’ ‘averaged,’ or ‘effective’ forces and rates, which reflect the behavior of the true forces and rates in the system at scales that are large with respect to \(\varepsilon \), while smoothing out the behavior at smaller scales. To derive an effective kinetic relation means to find a new relation between the limits of such macroscopic forces and rates as \(\varepsilon \rightarrow 0\), leading to a characterization of the kinetic relation for ‘the limiting system.’

Again, these effective kinetic relations are very common; for instance, Stokes’ law, Fourier’s law, Fick’s law, and many similar laws actually are effective kinetic relations, derived from more microscopic systems, often consisting of particles. Throughout science, such effective kinetic relations are the starting point for the modeling of dissipative systems at an effective scale [6, 23, 32, 33]. A detailed understanding of the properties and assumptions that lie at the basis of such effective kinetic relations is therefore essential.

We now return to the question of what we mean by a ‘healthy’ and an ‘unhealthy’ kinetic relation. The limiting dissipation potential \(\widetilde{{\mathcal {R}}}_0\) in the second example above depends on the energy \({{\mathcal {E}}}\), i.e., \(\widetilde{{\mathcal {R}}}_0(q,v) = {{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}(q)}(q,v)\). It follows that the contact set \({{\mathcal {C}}}_{\bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}\oplus \bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}^*}\) also depends on \({\mathrm D}{{\mathcal {E}}}\). The gradient flow equation (1.10) then takes the self-referential formThe induced evolution equation is correct, since different occurrences of \({\mathrm D}{{\mathcal {E}}}(q)\) interact nicely. However, the set \({{\mathcal {C}}}_{{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}\oplus {{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}^*}\) does not make sense as an independent kinetic relation, because \({{\mathcal {C}}}_{{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}\oplus {{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}^*}\) does not provide us with valid information about admissible pairs \((v,\xi )\) other than for the case \(\xi = -{\mathrm D}{{\mathcal {E}}}(q)\). In order to find the rate \(\dot{q}\) for a force \(\widehat{\xi }\ne -{\mathrm D}{{\mathcal {E}}}(q)\), we would need to construct a different energy \(\widehat{{\mathcal {E}}}(q)\) such that \(\widehat{\xi }= -{\mathrm D}\widehat{{\mathcal {E}}}(q)\), repeat the convergence process for this energy \(\widehat{{\mathcal {E}}}\), obtain a different limiting dissipation potential \(\widehat{{\mathcal {R}}}_{-{\mathrm D}\widehat{{\mathcal {E}}}}\), and read off the admissible rate \(\dot{q}\) from the resulting contact set \({{\mathcal {C}}}_{\widehat{{\mathcal {R}}}_{-{\mathrm D}\widehat{{\mathcal {E}}}}\oplus \widehat{{\mathcal {R}}}_{-{\mathrm D}\widehat{{\mathcal {E}}}}^*}\). Since this latter set is generically different from \({{\mathcal {C}}}_{\bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}\oplus \bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}^*}\), this shows how a single contact set \({{\mathcal {C}}}_{\bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}\oplus \bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}}^*}\) cannot be considered as a kinetic relation.

Instead, we seek a limiting kinetic relation that is defined as one single set \({{\mathcal {C}}}\) of pairs \((v,\xi )\) that provides us with all admissible combinations. The convergence concepts that we construct below are constructed with this aim in mind.

In the example of Sect. 1.3, the ‘correct’ effective dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}(q,v) = \bar{\mu }(q) v^2/2\) is obtained solely from information encoded in \({{\mathcal {R}}}_\varepsilon \). When considering a family of \(\Gamma \)-converging energies \({{\mathcal {E}}}_\varepsilon \xrightarrow {\Gamma }{{\mathcal {E}}}_0\), however, the ‘correct’ limiting dissipation potential may also contain information from \({{\mathcal {E}}}_\varepsilon \). This may seem to contradict our claim from above that the dependence of the effective dissipation on the energy is ‘unhealthy.’ As we shall see below, however, ‘correct’ or ‘healthy’ will mean that the effective dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}\) can depend on ‘microscopic details’ of \({{\mathcal {E}}}_\varepsilon \) but not on its ‘macroscopic limit’ \({{\mathcal {E}}}_0\). This expectation is stimulated by the idea of deriving a proper decomposition of ‘energy storage’ and ‘dissipation mechanisms’ in the macroscopic level.

To illustrate this, we revisit the classical example of a gradient flow in a ‘wiggly’ energy landscape [1, 10, 21, 34]. Again we take as state space \({\mathbf{Q}}={{\mathbb {R}}}\), but now the energy \({{\mathcal {E}}}_\varepsilon \) is \(\varepsilon \)-dependent while the dissipation potential \({{\mathcal {R}}}_\varepsilon ={{\mathcal {R}}}\) does not depend on \(\varepsilon \):where \({{\mathcal {E}}}_0:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is smooth and \(\varrho :{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) and \( A:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) are smooth and positive. The induced gradient flow evolution equation isIn Sect. 4, we summarize the results of [10] and place them in the context of this paper. We find that the system \(({{\mathbb {R}}},{{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}})\) converges in the simple EDP sense to a limiting system \(({{\mathbb {R}}},{{\mathcal {E}}}_0, \widetilde{{\mathcal {R}}}_0)\), where \({{\mathcal {E}}}_0\) is the \(\varepsilon \)-independent part of \({{\mathcal {E}}}_\varepsilon \) as in (1.11), and \(\widetilde{{\mathcal {R}}}_0\) is given bywhere this time the function \({{\mathcal {M}}}_0\) is given byAs in the previous example, \(\widetilde{{\mathcal {R}}}_0\) again depends on \({\mathrm D}{{\mathcal {E}}}_0(q)\). In Sect. 4, we also show that in the sense of one of the two new convergence concepts, namely contact EDP convergence with tilting, the family \(({{\mathbb {R}}},{{\mathcal {E}}}_\varepsilon , {{\mathcal {R}}})\) converges to a limiting system \(({{\mathbb {R}}},{{\mathcal {E}}}_0, {{\mathcal {R}}}_{\mathrm {eff}})\). Now, the effective dissipation potential \({{\mathcal {R}}}_{\mathrm {eff}}\) can be characterized explicitly viaWe see that \({{\mathcal {R}}}_{\mathrm {eff}}\) is independent of \({{\mathcal {E}}}_0\), but it depends on A, which is microscopic information contained in the family \({{\mathcal {E}}}_\varepsilon \). Moreover, the quadratic structure of \(v\mapsto {{\mathcal {R}}}(q,v)=\varrho (q)v^2/2\) is lost, because \({{\mathcal {R}}}_{\mathrm {eff}}(q,v)=|A(q)v|+ O(|v|^3)\) for \(v\rightarrow 0\).

The reason why gradient-flow convergence does not necessarily lead to a ‘healthy’ kinetic relation is relaxation: for a given macroscopic rate v and force \(\xi \), the limiting dissipation potential is found by a minimization over microscopic degrees of freedom constrained to the macroscopic imposed rate. This can be recognized in the definitions of \({{\mathcal {M}}}_0\) in (1.7b) and (1.12) and is very similar to the cell problems that arise in homogenization [8, 9, 19]. In the cases of this paper, the solutions of these cell problems may not be of gradient-flow type, leading to a situation where the limit problem does not describe a gradient-flow structure. We analyze this in more detail in Sect. 5.

To correct this, we introduce two novel aspects. The first is to consider not a single family \(({\mathbf{Q}},{{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )\) of gradient systems, but a full class of perturbed versions of this family. We perturb the given energies \({{\mathcal {E}}}_\varepsilon \) by arbitrary functions \({{\mathcal {F}}}\in {\mathrm C}^1({\mathbf{Q}})\):We call such a perturbation a ‘tilt’ and will then require convergence of all tilted systems simultaneously. The freedom to choose arbitrary tilts \({{\mathcal {F}}}\) allows us to probe the whole space of rates v and forces \(\xi \) for each q.

This setup leads to a first new convergence concept, which we call EDP convergence with tilting, or shortly tilt-EDP convergence. Unfortunately, it may suffer from the same problems of relaxation, and therefore, it is a rather restrictive concept that is too strong to cover the simple cases of wiggly dissipation and wiggly energy discussed above.

The second new aspect is to weaken the definition of tilt-EDP convergence to require only a reduced connection between the relaxed problem and the limiting dissipation potential—a connection that only holds ‘at the contact set \({{\mathcal {C}}}\).’ This leads to the concept of contact-EDP convergence with tilting, or shortly contact-EDP convergence. We show in the examples later in this paper that the concept of contact-EDP convergence for gradient systems yields kinetic relations that do not suffer from the force dependence that we observed above for simple EDP convergence.

In Section 2, we define gradient systems and gradient flows, recall the existing concept of simple EDP convergence, and introduce the two novel convergence concepts tilt-EDP convergence and contact-EDP convergence. These notions were already introduced in [10], but called ‘strict EDP convergence’ and ‘relaxed EDP convergence,’ respectively. In Sects. 3 and 4, we study in detail the examples of a wiggly dissipation potential and a wiggly energy, respectively, that were briefly mentioned above. In Sect. 5, we discuss in depth the reasons why the concept of contact-EDP convergence is an improvement over the classical concept of EDP convergence, and why it corrects the ‘incorrect’ kinetic relationship that we mentioned above.

In Sect. 6, we connect the tilting of energies as described above with tilting of random variables in large deviation principles and show how the independence of the dissipation potential from the force arises naturally in that context.

In Sect. 7, we present a result on tilt-EDP convergence that was formally derived in [22] and is rigorously treated in [17]. It concerns diffusion through a membrane in the limit of vanishing thickness and shows that even in the case of tilt-EDP convergence we can start with quadratic dissipation potentials \({{\mathcal {R}}}_\varepsilon (q,\cdot )\), i.e., linear kinetic relations, and end up with a non-quadratic effective dissipation potential, i.e., a nonlinear effective kinetic relation.

The context for this paper is a smooth finite-dimensional Riemannian manifold \({\mathbf{Q}}\), which may be compact or not. A common choice is \({\mathbf{Q}}= {{\mathbb {R}}}^n\). We write \(|\cdot |\) for the local norms on the tangent and cotangent spaces \({\mathrm T}{\mathbf{Q}}\) and \({\mathrm T}^*{\mathbf{Q}}\), and \({\mathrm T}{\mathbf{Q}}\oplus {\mathrm T}^*{\mathbf{Q}}\) for their direct (Whitney) sum

The dissipation potential has a natural Legendre–Fenchel dual \({{\mathcal {R}}}^*:{\mathrm T}^*{\mathbf{Q}}\rightarrow [0, \infty ] \),By our assumptions on \({{\mathcal {R}}}\), the dual potential \({{\mathcal {R}}}^*\) is also convex, lower semicontinuous, non-negative, and satisfies \({{\mathcal {R}}}^*(q,0)=0\). We denote the (convex) subdifferentials of \({{\mathcal {R}}}\) and \({{\mathcal {R}}}^*\) with respect to their second arguments as \(\partial _v{{\mathcal {R}}}\) and \(\partial _\xi {{\mathcal {R}}}^*\).

The following lemma gives a well-known connection between growth and subdifferentials:

The gradient-flow equation induced by the gradient system is, in three equivalent forms, The final line can be used to generate an additional formulation. For absolutely continuous curves \(q:[0,T]\rightarrow {\mathbf{Q}}\), in short \(q\in \mathrm {AC}([0,T],{\mathbf{Q}})\), define the dissipation functional asBy integrating the Young–Fenchel inequalitywith \(\xi = - {\mathrm D}{{\mathcal {E}}}(q)\) and using the chain rule, we find

On the other hand, by integrating (2.2c) in time we find that solutions q of (2.2) achieve equality in (2.5). This leads to a further characterization of solutions; see [4] or [27, Thm. 3.1]:

The energy-dissipation principle formulation (2.6) of a gradient flow leads to a natural concept of gradient system convergence. A first version of this concept was formulated by Sandier and Serfaty [36] and generalizations have been used in a large number of proofs (see, e.g., [5, 22, 24‐26, 29, 39]).

The two parts of Definition 2.7 naturally combine to enable passing to the limit in the integrated formulation (2.6), as illustrated by the proof of the following lemma.

In the definition of simple EDP convergence, as well as in the two versions of EDP convergence with tilting, we ask for the full \(\Gamma \)-convergences \({{\mathcal {E}}}_\varepsilon {\mathop {\longrightarrow }\limits ^{\Gamma }}{{\mathcal {E}}}_0\) and \({\mathfrak {D}}^T_\varepsilon {\mathop {\longrightarrow }\limits ^{\Gamma }}{\mathfrak {D}}^T_0\). This is needed to define the limits \({{\mathcal {E}}}_0\) and \(\widetilde{{\mathcal {R}}}_0\) in a unique way. For studying the limiting solutions \(q_0\) as in Lemma 2.8, the two liminf estimates are enough; however, our aim is to recover effective kinetic relations or effective dissipation potentials, which is additional information not contained in the limit equation.

Also in the fundamental work [36, 39] on evolutionary \(\Gamma \)-convergence for gradient flows only the liminf estimates are imposed, because there the main focus is on the characterization of the limit solutions.

As we explained in Introduction, simple EDP convergence may lead to ‘unhealthy’ limiting dissipation potentials, which violate the requirement (1.8)–(1.9). As a central step toward improving the situation, we embed the single sequence \(({\mathbf{Q}},{{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )\) in a family of sequences \(({\mathbf{Q}},{{\mathcal {E}}}_\varepsilon {+} {{\mathcal {F}}},{{\mathcal {R}}}_\varepsilon )\), parameterized by functionals \({{\mathcal {F}}}\in {\mathrm C}^1({\mathbf{Q}};{{\mathbb {R}}})\), thereby ‘tilting’ the functionals \({{\mathcal {E}}}_\varepsilon \). Tilting \({{\mathcal {E}}}_\varepsilon \) does not change the \(\Gamma \)-convergence properties: we haveHowever, for the dissipation functional \({\mathfrak {D}}_\varepsilon ^T\) we obtain new and nontrivial information by considering the dissipation functional for the tilted energy:We now assume that the \(\Gamma \)-limits of \({\mathfrak {D}}_\varepsilon (\cdot ,{{\mathcal {F}}})\) exist, i.e.To recover the original structure of integrals \({\mathfrak {D}}_\varepsilon ^T\) in terms of \({{\mathcal {M}}}_\varepsilon \), we definesuch that \({\mathfrak {D}}_0^T\) has the desired formWe capture this discussion in a definition that provides the basis for the later convergence concepts.

We briefly comment on these. Assumptions 1–3 make the prior discussion precise. Note that \({{\mathcal {N}}}_0\) is assumed to be independent of the time horizon T. This is a common feature of convergence results of this type; see, e.g. [8, Ch. 3], or the examples later in this paper, and note that this independence is also implicitly present in condition (2.7) for simple EDP convergence.

Assumption 4 is the expected consequence of the Fenchel-Young inequality (2.4) giving \({{\mathcal {M}}}_\varepsilon (q,v,\xi ) \ge \langle \xi ,v\rangle \) for all \(\varepsilon >0\). This assumption is needed to obtain the upper energy estimate (2.5) for the limit functionals \({{\mathcal {E}}}_0\) and \({\mathfrak {D}}^T_0\) as well, namelyAssumption 5 is satisfied at positive \(\varepsilon \), since by the conditions on dissipation potentials we have \({{\mathcal {R}}}_\varepsilon (q,v)\ge {{\mathcal {R}}}_\varepsilon (q,0)=0 \) for all q and v, so thatSince the property \({{\mathcal {R}}}_\varepsilon (q,v)\ge {{\mathcal {R}}}_\varepsilon (q,0) = 0 \) is an intrinsic property of gradient systems (see Remark 2.6), Assumption 5 formulates that the limiting structure \({{\mathcal {M}}}_0\) preserves this aspect of the gradient-flow nature. If we impose a continuity requirement on \({{\mathcal {N}}}_0\), then Assumption 5 can also be derived through the \(\Gamma \)-convergence limit—we show this in the next lemma. In the next section, both Assumptions 4 and 5 will be essential in recovering a dissipation potential formulation of \({{\mathcal {M}}}_0\).

For fixed \(q\in {\mathbf{Q}}\), the map \( (v,\xi )\mapsto {{\mathcal {M}}}_0(q,v,\xi )\) constructed in the previous section may have various different properties, and we study them next.

Let X be a real reflexive Banach space; we will apply the results below to the case \(X={\mathrm T}_q{\mathbf{Q}}\) and \(X^* = {\mathrm T}_q^*{\mathbf{Q}}\), for a fixed \(q\in {\mathbf{Q}}\), but the development below holds more generally. Recall that any functional \({{\mathcal {R}}}:X\rightarrow [0,\infty ]\) is a dissipation potential if it is convex, lower semicontinuous, non-negative, and satisfies \({{\mathcal {R}}}(0)=0\).

We now define two new convergence concepts, EDP convergence with tilting and contact EDP convergence with tilting.

We add a statement on simple EDP convergence for completeness and comparison:

In each of the three cases, the convergence uniquely fixes a limiting dissipation potential \(\widehat{{\mathcal {R}}}_0(q,\cdot )\), \({{\mathcal {R}}}_{\mathrm {eff}}(q,\cdot )\), or \(\widetilde{{\mathcal {R}}}_0(q,\cdot )\) for tilt-EDP, contact-EDP, or simple EDP convergence.

In Section 1.4, we described how we want the new convergence concepts to be such that tilting the energies does not change the effective dissipation potentials. The definitions above have been constructed with this aim in mind, and we now check that indeed the two tilted convergence concepts have this property.

Next, we consider relations between the three convergence concepts. Up to a technical requirement, the three concepts are ordered:

The proof directly follows by reshuffling the definitions.

The important thing to note here is that the problems with simple EDP convergence, in having force-dependent dissipation potentials, cannot be solved simply by requiring simple EDP convergence for all tilted versions of the systems with a single dissipation potential. By Lemma 2.19, this requirement is equivalent to tilt-EDP convergence and therefore is too strong: in the two examples of Sects. 3 and 4, tilt-EDP convergence does not hold.

The benefit of the intermediate concept of contact-EDP convergence lies in the combination of tilting, which allows the convergence to roam over all of \((v,\xi )\)-space, with restriction to the contact set, which allows the connection between \({{\mathcal {M}}}_0\) and \({{\mathcal {R}}}_0\) to focus on the case of contact, i.e., the kinetic relation. We comment more on this in Sect. 5.

We study a family \(({{\mathbb {R}}},{{\mathcal {E}}},{{\mathcal {R}}}_\varepsilon )\), \(\varepsilon >0\), of gradient systems, where the energy is independent of \(\varepsilon \) while the dissipation strongly oscillates in the state variable q, namelywhere \(\mu \in {\mathrm C}^0({{\mathbb {R}}}^2)\) is 1-periodic in the second variable, i.e., \(\mu (q,y{+}1)=\mu (q,y)\), and has positive lower and upper bound \(0< {\underline{m}}\le \mu (q,y)\le \bar{m}<\infty \). We setCombining Theorem 3.1 and Lemma 2.8, we obtain the following convergence result for the gradient-flow equations. The solutions \(q^\varepsilon \) ofconverge to the solution q of the gradient flow

We emphasize that the gradient-flow equation obtained from simple EDP convergence is indeed the same as the equation obtained from contact-EDP convergence:This form can be more explicit by using the fact that \({{\mathcal {M}}}_0(q,\cdot ,\cdot )\) only depends on \(v^2\) and \(\xi ^2\) and is homogeneous of degree one in these variables, viz.This follows from the explicit representation of \({{\mathcal {M}}}_0\) given in (3.3c). The function \(\Phi :{{\mathbb {R}}}\times [0,1]\rightarrow {{\mathbb {R}}}\) is continuous and satisfieswhere the last relation follows from \({{\mathcal {M}}}_0(q,v,\xi )\ge \xi v\). With this, we find the force-dependent dissipation potentialand with \(\widetilde{{\mathcal {R}}}_0(q,v)=\bar{{\mathcal {R}}}_{-{\mathrm D}{{\mathcal {E}}}(q)}(q,v)\) the gradient-flow equation (3.2) takes the formUsing \(\partial _s\Phi (q,1/2)=0\), we have \(\Psi (q,1/2)=\Phi (q,1/2)=1/(2\bar{\mu }(q))\) and conclude that (3.2) is indeed equivalent to (3.1).

Certainly, this form of the equation involving the nonlinear kinetic relationis ‘unhealthy’ in the sense discussed above; in particular, it is ‘less natural’ than the effective equation (3.1) featuring the simple linear kinetic relation \(v \mapsto \xi =\bar{\mu }(q)v\).

Here we prove the EDP convergences stated above.

We discuss a few specific points for this model that complement the results in [10] for the wiggly energy model to be discussed in the following section.

In [29], we showed the following general result: Suppose that \(Q^n\) is a sequence of continuous-time Markov processes in \({\mathbf{Q}}\) that are reversible with respect to their stationary measures \(\mu ^n\in {\mathcal {P}}({\mathbf{Q}})\). Assume that the following two large deviation principles hold: Then, I can be written asfor some symmetric dissipation potential \({{\mathcal {R}}}\). This result can be interpreted as follows.Over the last few years, a number of well-known gradient systems have been recognized as arising in this way. For instance, the ‘diffusion’ or ‘heat’ equation \(\partial _t \rho = \Delta \rho \) arises as the limit of independent (‘diffusing’) Brownian particles [2, 3], with the well-known entropic Otto–Wasserstein gradient structure (cf. [31] and our Sect. 7); as the limit of the simple symmetric exclusion process describing particles hopping on a lattice [3], with a gradient structure of a mixing entropy and a modified Wasserstein distance; and as the limit of oscillators that exchange energy (‘heat’) [35], with a gradient structure consisting of an alternative logarithmic entropy and again a modified Wasserstein distance. Rate-independent systems arise from taking further limits [7], and extensions to GENERIC have also been recognized [12].

We first consider a non-dynamic case: \(X^n\) is a random variable in \({\mathbf{Q}}\), with law \(\mu ^n\in {\mathcal {P}}({\mathbf{Q}})\). One example of this arises in the stochastic process example above: if the initial state \(Q_0^n\) is drawn from the invariant measure \(\mu ^n\) of the process, then \(Q^n_t\) also has law \(\mu ^n\) for all time \(t\ge 0\), and \(X^n :=Q^n_{t}\) for fixed t therefore is an example of the situation we are considering.

In previous sections, we have implicitly used a property that is well known in the context of energetic modeling: Energies are additive. More precisely, when combining energies that arise from different phenomena, the energy of the total system is simply the sum of the individual energies. In this way, given an energy \({{\mathcal {E}}}\), the perturbed energy \({{\mathcal {E}}}+{{\mathcal {F}}}\) arises naturally as the sum of the original energy \({{\mathcal {E}}}\) and the external potential \({{\mathcal {F}}}\).

We now connect this additivity property with tilting of random variables. In the stochastic context, tilting a sequence of random variables \(X^n\) means considering a new sequence \(X^{{{\mathcal {F}}},n}\) with lawThis has the effect of giving higher probability to \(q\in {\mathbf{Q}}\) for which \({{\mathcal {F}}}(q)\) is smaller: it ‘tilts’ the distribution in the direction of lower values of \({{\mathcal {F}}}\).

If \(\mu ^n\) satisfies a large deviation principle with rate function S, as in the case of the stochastic process above, and satisfies a tail condition, then Varadhan’s and Bryc’s Lemmas (see, e.g., [13, Th. II.7.2]) imply that \(\mu ^{{{\mathcal {F}}},n}\) also satisfies a large deviation principle, with ‘tilted’ rate function \(S^{{\mathcal {F}}}\):where the constant is chosen such that \(\inf S^{{\mathcal {F}}}= 0\). This result can be understood by remarking that from \(\mu ^n\sim {\mathrm e}^{-nS}\) we findwhich leads to the first two terms in \(S^{{{\mathcal {F}}}}\); the constant in \(S^{{\mathcal {F}}}\) arises from the normalization constant in (6.3).

The additivity property for energies thus has a counterpart for random variables in the form of the tilting of (6.3); the two concepts, addition of energies and tilting of random variables, coincide in the large deviation limit \(n\rightarrow \infty \).

In the setup in the previous sections, not only are energies assumed to be additive, but also the dissipation function \({{\mathcal {R}}}\) is assumed to be independent of the tilting: the addition of \({{\mathcal {F}}}\) changes the energy but not the dissipation. This assumption has its origin in the modeling background of mechanical gradient flows, in which the dissipation functional \({{\mathcal {R}}}\) defines the force-to-rate relationship \({\mathrm D}_\xi {{\mathcal {R}}}^*(q,\cdot )\), which is assumed to be independent of the driving energy.

We now show that the same independence occurs naturally for gradient systems that arise in the context of Markov processes. As in Sect. 6.1, we consider a Markov process \(Q^n\) in \({\mathbf{Q}}\) with generator \(L^n\). (For instance, if \(Q^n\) solves the stochastic differential equation in \({{\mathbb {R}}}^d\),thenIn the dynamic context, tilting can be written in terms of the generator through the Fleming–Sheu logarithmic transform [15, 40],Let \(Q^{{{\mathcal {F}}},n}\) be generated by \(L^{{{\mathcal {F}}},n}\); if \(Q^n\) has invariant measure \(\mu ^n\), then \(Q^{{{\mathcal {F}}},n}\) has the invariant measure \(\frac{1}{Z_n}\, {\mathrm e}^{-n{{\mathcal {F}}}}\mu ^n\) with \(Z_n:=\int _{\mathbf{Q}}{\mathrm e}^{-n{{\mathcal {F}}}}\;\!\mathrm {d}\mu ^n\).

In the derivation of the characterization (6.2), \({{\mathcal {R}}}^*\) is found by taking the limit in a scaled version of \(L^n\), as follows. Define the nonlinear generatorand its limit, in a sense to be defined precisely (see [14, Ch. 6, 7]),In a successful large deviation result, the operator H operates on f only through its derivative \({\mathrm D}f\), which allows us to identifyThe dual dissipation function \({{\mathcal {R}}}^*\) is then defined byGiven this structure, we can now show how tilting does not affect \({{\mathcal {R}}}^*\). If we replace \(L^n\) by \(L^{{{\mathcal {F}}},n}\) in this procedure, thenThe dissipation potential \({{\mathcal {R}}}^{{{\mathcal {F}}},*}\) associated with the large deviations of the tilted process \(Q^{{{\mathcal {F}}},n}\), with tilted invariant-measure rate functional \(S^{{\mathcal {F}}}= S +{{\mathcal {F}}}+ \mathrm {constant}\), then satisfiesIn other words, tilting replaces the invariant-measure large deviation functional S by \(S^{{\mathcal {F}}}= S+ {{\mathcal {F}}}+\mathrm {constant}\) and leaves \({{\mathcal {R}}}\) untouched.

Summarizing, there is a strong analogy between the modification of energies by addition and the modification of stochastic processes by tilting. In both cases, the dissipation function is expected to be unaffected; in the mechanical context this is a modeling postulate, and in the stochastic context it is a consequence of the structure of the tilting.

Regardless of whether the gradient-flow structure arises directly from a modeling argument or indirectly through a large deviation principle, the behavior under modification of the energy is therefore the same.