110th Anniversary: Mesoscale Complexity—To Dodge or To Confront

Commentary pubs.acs.org/IECR

Cite This: Ind. Eng. Chem. Res. 2019, 58, 12478−12484

110th Anniversary: Mesoscale ComplexityTo Dodge or To Confront? Wen Lai Huang,† Jinghai Li,*,† and Xiaosong Chen‡ †

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State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China ‡ School of Systems Science, Beijing Normal University, Beijing, 100875, People’s Republic of China ABSTRACT: This commentary briefly addresses the complexity at mesoscales of different levels, describes the limitations of traditional approaches, and introduces the preliminary principle of the potential mesoscience. It is believed that mesoscale complexity arises from collective effects, and emerges within mesoregimes. Such complexity makes the averaging approaches (based on the assumption of homogeneity) solely at the system scale (or within each computational grid) problematic while the ab initio approaches solely at the element scale are commonly unaffordable to achieving system behavior. The difficulty in finding a single-objective extremal principle and the invalidity of the homogeneity assumption indicates that the equilibrium framework might not be a good starting point to understand the behavior of systems far from equilibrium. Focusing on the specific degrees of freedom at mesoscales and revealing the corresponding multiple dominant mechanisms seemingly pave a promising way, and this methodology leads to the concept of mesoscience. Some topics closely related to chemistry have been proposed within such a framework as well.

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INTRODUCTION

The fact is that we cannot (probably never) afford addressing every issue (actually the behavior of a system) from its very roots (say, at the element scale at the involved lowest level) directly and solely, especially when we handle complex systems where many-body long-range interactions are significant. To address, not to neglect (approaches solely at the system scale (macroscale) might do) the complexity, it is usually indispensable to bridge the huge spatiotemporal gaps between the element scales at the involved lowest levels and the system scales at the involved highest levels, and thus we commonly need multilevel (when multiple levels are involved) multiscale approaches where mesoscales at different levels call for special attention.1,7,8

Complex systems are difficult to understand, describe, and manipulate. The challenging complexity there results from the prevalence of many-body, usually long-range, interactions among their elements, and the massive exchange of mass and energy between the systems and their environments through their boundaries.1 To tackle such complexity, great efforts have been made for a long time. However, no triumph can be declared yet, and the exploration footprints themselves are very complex, as reflected in the map2 showing the development of complexity sciences. Horgan even stated that complexity science itself is still of perplexity,3 signaling the underlying gigantic challenge. Within the field of chemistry, such complexity prevails as well, which greatly limits the applicability of ab initio approaches at the element scale (microscale). Just after the proposition of the Schrödinger equation, in 1929 Dirac remarked that “the fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known”.4 Seemingly, in the words of Kohn,5 “chemistry had come to an end”. However, meanwhile, Dirac realized that in almost all cases,5 “application of these laws leads to equations that are too complex to be solved”,4 signifying the despair for the practical value of such approaches. Therefore, not surprisingly, chemistry continued and still continues to flourish, accompanied by many (e.g., multiscale6) approaches more applicable to describing system behavior than the exact and sole Schrödinger equation, including the remarkable achievements4,5 in solving the Schrödinger equation. © 2019 American Chemical Society

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TRADITIONAL APPROACHES AND EQUILIBRIUM FRAMEWORK The efforts in describing the behavior of complex systems can be classified into three categories. The first category is at the element scale, using the constitutive relationships and conservation laws at that scale. Depending on the nature (classical or quantum) of the elements, Newtonian9 or quantum mechanics4 will be adopted. The difficulty lies in the fact that based on Newtonian mechanics we are not capable of analytically describing the behavior of even three interacting bodies10 till today. Through numerical approximation, an increasing number of objects can be handled affordably with increasing computational capacity, but the affordable system size is still quite limited (far below the practical size in most cases) nowadays. For example, current Published: June 21, 2019 12478

DOI: 10.1021/acs.iecr.9b01655 Ind. Eng. Chem. Res. 2019, 58, 12478−12484

Commentary

Industrial & Engineering Chemistry Research state-of-the-art11−13 simulations of classical molecular dynamics14 usually treat with just around billions of atoms, far below one mole, and within only very limited time intervals, say, nanoseconds. Therefore, it seems still safe to say that employing Newtonian mechanics solely at the element scale is unaffordable to capture the behavior of a system with a practical number (say, around or above one mole) of elements for a practical time period (say, around or above one second) for most systems even nowadays, although it is capable in principle. The situation in performing computations based on quantum mechanics is more forbidding even when many approximations are adopted.4,5 This happens because at the element scale we have too many degrees of freedom (DoFs) to track. Although most of the DoFs at the element scale might be insignificant or even useless, and thus redundant for describing the system behavior, we cannot distinguish or screen them at the element scale. Therefore, due to computational cost, the application of this category of approaches is normally limited to special systems (with a small number of elements) under special cases (within very short time period). The second category is at the system scale, using the constitutive relationships (or principles) and conservation laws at that scale, where nonequilibrium thermodynamics15 is a well-known example. Since working solely at the system scale, not concerning the details at smaller scales, these approaches hardly give sufficient consideration to the complexity. Actually, averaging based on the homogeneity assumption is frequently adopted, at least within each computational grid, possibly leading to severe deviations or even unphysical results. As to nonequilibrium thermodynamics, there is an additional difficulty in finding the extremal principle to be adopted. As we know, revealing the common applicable principle in the nonlinear regime seems still very thorny,16 and a unanimous solution has not been achieved.17 Various single-objective extremal principles have been proposed, showing a fragmented landscape15 with debates,18,19 so there is still a lack of full confidence when adopting them. Both the assumption of homogeneity to facilitate averaging and the attempt to find a single-objective extremal principle might have some roots in the framework of equilibrium thermodynamics,20 so it might be useful to revisit the equilibrium framework, revealing its limitation in tackling complexity. Classical thermodynamics describes equilibrium systems directly at the system scale. On the basis of experiments, equilibrium assumptions, and inference, thermodynamic laws and several system quantities (e.g., thermodynamic entropy) were established for the system behavior, that is, at the system scale, without concern about the details on the element scale. Afterward, such laws and relevant system quantities (temperature, pressure, entropy, etc.) were further interpreted from the kinetic theory of gases and classical statistical mechanics, that is, at the element scale, but the correlation between the system scale and the element scale was based on the assumption of equilibrium, that is, ergodicity,20 which enables the simplification of averaging. By the way, in those days, the atomic theory remained unestablished.21 Therefore, it is clear that classical thermodynamics is a theory at the system scale, and its interpretation by classical statistical mechanics reveals that it is actually a theory for infinite systems, that is, infinite elements or infinite time, for ergodicity to hold and thus equilibrium to arrive. Without such a prerequisite, for example, in finite systems (even small systems), local space or time, or low-dimensional systems, the laws of classical

thermodynamics and relevant concepts (e.g., temperature) will be questionable. For instance, the equivalent (or extension) of the second law of classical thermodynamics can be violated in small systems. (What is violated is not the classical thermodynamics, but its prerequisites.) The Clausius statement of the second law is that “heat can never pass from a colder to a warmer body without some other change”.22 However, in local space and time, say, just observing several times of collisions of two molecules, we may find that the kinetic energy can transfer from a molecule with lower kinetic energy to a molecule with higher kinetic energy, that is, the transfer reduces the kinetic energy of the low-energy molecule, but enhances the kinetic energy of the high-energy molecule. A simple example can be given in a two-dimensional perfectly elastic collision of two monatomic molecules (#1 and #2) with identical mass. Suppose that molecule #1 has a zero y-component but molecule #2 has a finite y-component (defining y along the line connecting the centers of the two molecules at collision), after the collision, molecule #1 will have a finite y-component while the ycomponent of molecule #2 will be zero. If the molecule #1 has a larger x-component than molecule #2, then such a collision will transfer the kinetic energy from the low-energy molecule #2 to the high-energy molecule #1. If we reckon that kinetic energy represents temperature (in an average flavor), the above observation seems to violate the second law (locally and instantaneously). Again, when observing an isolated small system containing only two molecules within a short time interval, considering the occupancy of two predefined sides (say, separated by an imaginary plane), we may find that the homogeneous state (each molecule occupies a different side) does not exist longer than any segregating state (two molecules occupy one definite side). Although the homogeneous state corresponds to more configurations (i.e., a larger multiplicity factor16), and thus higher entropy, this state may appear less frequently than a segregating state within a specific time interval. Another example is the one-dimensional perfectly elastic collisions of identical spheres within smooth boundaries (Newton’s cradle23 is a special case), which cannot yield an equilibrium state showing the Boltzmann distribution20 of the particle (spheres can be taken as monatomic molecules) velocities since ergodicity cannot be realized (the velocity distribution keeps fixed at its initial one). In other words, the ergodicity assumed in infinite systems of classical thermodynamics does not hold in lots of practical circumstances, for example, small systems. When a system is small enough, that is, the number of elements is small enough and the observing time is short enough, the number of forward trajectories is comparable with that of backward trajectories,24 and the probability of homogenizing is comparable with that of heterogenizing, exhibiting the reversibility of homogenizing or heterogenizing. With the increase in the system size (the number of elements or the observing time), the multiplicity factor for the homogeneous state increases rapidly,16 and thus the number of trajectories toward the homogeneous state, but not the number of trajectories toward heterogeneous (segregating) states. Therefore, the probability of homogenizing ascends significantly, showing the increasing irreversibility of homogenizing.24 In this aspect, the fluctuation theorem offers valuable statistical descriptions,24 but there is still a long way to complete understanding and rational manipulation of small systems, and it seems important to determine the boundary and describe the transition between small systems and infinite systems. 12479

DOI: 10.1021/acs.iecr.9b01655 Ind. Eng. Chem. Res. 2019, 58, 12478−12484

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Industrial & Engineering Chemistry Research

the single-objective extremal principle has become such a charming dream that attempts to find it have never stopped. However, when we attempt to tackle the complexity, systematic thinking is advocated, and multiobjective ideas seem valuable. In other words, the equilibrium framework cannot help much (if not helpless at all, or even give hindrance due to its deep influence on thinking) in developing theories for systems far from equilibrium or finite systems. This is due to the distinction between equilibrium systems and other systems, and the equilibrium framework was established strictly on the unique (actually very special and ideal16) nature (ergodicity) of equilibrium systems, which cannot be borrowed readily to other systems. In this circumstance, it is better to abandon the equilibrium framework completely and search for a (almost) totally different framework to describe complex systems if possible.

When ergodicity is fulfilled, averaging based on the homogeneity assumption is reasonable. However, when ergodicity is violated, such averaging is problematic. As to the averaging approach, it is well known that Landau proposed the general description of mean-field (MF) approaches in 1937.25 Soon afterward, it was found that MF approaches fail completely in describing critical phenomena. After decades of efforts, Ginzburg arrived at two key findings. The first is that the MF approach is valid if the mean amplitude of thermal fluctuations is less than the amplitude of the observables, and the second is that the MF description gives the correct critical behavior for systems with spatial dimension equal to or greater than four, the wellknown Ginzburg’s criterion.25 This criterion was verified subsequently by Wilson in his theory of renormalization group.25 For complex systems, fluctuations are usually dramatic, and critical regimes are normally encountered, so MF-based averaging can lead to dramatic errors. For instance, in gas−solid fluidization, the deviation can reach several orders of magnitude,26,27 and significant errors have also been observed in heterogeneous catalysis.28−31 The aim of developing nonequilibrium thermodynamics, extended from the classical equilibrium thermodynamics, may be to explain the various ordering (self-organization) behavior in the real world, for example, the spectacular life systems, which cannot be understood in the classical equilibrium thermodynamics. Such ordering may be classified into two categories. One belongs to the dissipative structures defined by Prigogine.32 Such systems are still of the size of classical thermodynamic systems, that is, with sufficient number of elements and sufficient period of time. Even the spatial local points in them can still be viewed to be “macroscopically infinitesimal but microscopically infinite”, and the local equilibrium assumption is usually adopted. The other comes from small systems; for example, an adult human being contains only around 1014 (far below one mole) cells that are the functional elements of the human system. Small systems are intrinsically in nonequilibrium, since the equilibrium quantities defined in classical thermodynamics cannot be well-defined here. However, there is no definite theory for thermodynamic systems (large enough) far from equilibrium so far, let alone that for small systems. Perhaps the specificity of small systems is the main cause to the flourishing of the nanometer community where there are lots of interesting things and also lots of difficulties to understand and manipulate, since we have on hand (relatively complete and reliable) only classical thermodynamics and classical statistical mechanics for infinite equilibrium systems, and classical mechanics and quantum mechanics to deal with elements. Since ergodicity is not easily fulfilled in small systems, adopting the theory, approach, and concepts of classical thermodynamics therein is usually infeasible. Therefore, abnormal results might emerge when we try to apply free energy, temperature, etc. in nanometer systems (with the system scale around nanometers, and the element of atoms or molecules), indicating that the classical equilibrium framework does not fit there anymore. The other difficulty in attempting to develop theories at the systems scale for systems far from equilibrium or finite systems, that is, complex systems, through revising or extending the theories of equilibrium systems (under the assumption of ergodicity), is possibly due to searching for a single-objective extremal principle similar to those (maximization of entropy or minimization of free energy) in equilibrium thermodynamics. Possibly since it works so well in equilibrium thermodynamics,

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MULTIOBJECTIVE VARIATIONAL APPROACH AND MESOSCIENCE Now let us turn to the third category of approaches in dealing with complex systems. This category respects the multiscale nature of a complex system, recognizing that there exists the key and mostly critical mesoscale behavior between the element scale and the system scale. However, establishing the constitutive relationships at the mesoscale is very difficult, so three typical approaches emerge with different strategies.1,7 The first approach is descriptive. It describes the mesoscale structures, but usually does not establish the constitutive relationships, so its value is commonly limited within characterizing the structures. The second approach is correlative. It establishes the constitutive relationships via fitting data, so the transferability is normally limited. The last approach is multiobjective variational, an example of which is the energyminimization multiscale (EMMS) approach.33,34 Here the multiobjective variational principles at the mesoscales are pursued instead of the constitutive relationships. The EMMS principle, that is, the principle of compromise in competition1,7,33,34 has seemed promisingly valid, as evidenced in mesoscale modeling for several systems,1,7,28,33−36 and the difficulty of pursuing the constitutive relationships at the mesoscale might be avoided. On the basis of the EMMS principle, mesoscience1,7,8,37−39 was proposed to deal with the inherent complexity in complex systems, featuring spatiotemporal heterogeneity, from a viewpoint totally different from those of other approaches. In mesoscience, analysis of a system will be started with identifying the level it belongs to, and defining quantities at its element scale, mesoscale, and system scale, respectively. Then, we focus on the mesoscale, revealing the inherent dominant mechanisms that govern the complexity. Since each dominant mechanism corresponds to a respective variational expression, the coexistence (via compromise in competition) of multiple dominant mechanisms yields a multiobjective variational formulation40−42 (though a single-objective expression is possible through further physical or mathematical derivation). Therefore, quantities at different scales can be correlated by conservation laws and variational principles. This framework adopts no presumption as that in the theories for equilibrium systems, so it avoids the distinction between equilibrium and nonequilibrium (and the distinction between linear and nonlinear regimes of nonequilibrium), and hence the limitation due to such a distinction (or distinctions). Its resolution of scales and focus on the mesoscale distinguishes it naturally from the 12480

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Industrial & Engineering Chemistry Research theories at the system scale and those at the element scale. Its applicability does not put requirements on the system size (i.e., applicable to both infinite and finite systems), and meanwhile it does not involve the redundant DoFs at the element scale. Actually, for different systems (distinguished on the basis of traditional concepts, e.g., nonequilibrium, equilibrium; linear, nonlinear; infinite, finite, small; conservative, dissipative) we can all find the compromise in competition between different dominant mechanisms, which suggests that the EMMS principle might lay a basis for a more general framework. Mesoscience is based on the assumption that the DoFs at the mesoscale of complex systems are different from those at the element scale. Owing to collective effects, huge numbers of DoFs at the element scale degenerate or combine at the mesoscale and therefore lead to significantly reduced DoFs at the mesoscale. In other words, the DoFs related to elements are simplified to much fewer DoFs related to mesoscale structures due to the collective behavior of elements. Therefore, it is reasonable (and more efficient) to work in a phase space with reduced dimensions (concerning the generalized coordinates and momenta of the mesoscale structures). Such an assumption might be reflected somewhat by the fact that for dissipative systems, their volume in phase space (spanned by the generalized coordinates and momenta of all elements) is not conserved along the evolution trajectories,24 indicating that transient or steady states exhibit different DoFs from the ergodic state, and the phase space for equilibrium systems might be redundant for complex systems. Within such a framework, it has been recognized that all spatiotemporal structures result from compromise in competition between at least two dominant mechanisms, each of which gives a respective variational expression (possibly following a respective variational principle). The fluctuation of structures is related to the alternating dominance between these dominant mechanisms. In addition, the dominance of each mechanism can change with the operating regime. For a system with two dominant mechanisms (e.g., A and B), three typical regimes can emerge successively with the change in the relative dominance between A and B, that is, A-dominated, A−B compromise-incompetition, and B-dominated regimes, and spatiotemporal structures emerge within the mesoregime (the A−B compromise-in-competition regime).8,38,39,43,44 With these understandings, the following points seem evident:

processes or mechanisms,39 and therefore, the extremal feature of dissipative processes is process- or mechanismspecific, such as minimum for suspension of solids but maximum for other dissipative processes in gas−solid fluidization systems, and different extremal tendencies for viscosity-dominated and inertia-dominated processes in turbulence. In thermodynamics, the total entropy production blurred all dissipation into a total, leading to the loss of the process-specific information, and hence, giving rise to the difficulty in defining the variational functionals in mesoregimes at different levels.8 This is likely the reason why debate and difficulty exists in thermodynamics for mesoregimes where at least two dissipative processes occur. Prigogine’s concept of dissipative structure is revolutionary, proposing the notion that nonequilibrium is the origin of order, but constrained, unfortunately, in looking for a single variational functional related to entropy. • Dominant mechanisms are level- and system-specific: Although dominant mechanisms show possible correlation with the extremal tendencies of total energy dissipation rates in several systems,44 which might provide useful clues for extracting them, the actual expressions of dominant mechanisms might not all be extremal tendencies of the total energy dissipation rates, but level-specific (relevant to the level in question) and even system-specific (relevant to the exact processes in the system). In our experience, extremal tendencies of terms concerning structural quantities, rates of energy consumption (part), or rates of energy dissipation (total or part) could be employed.44 • Limitation of homogeneity presumption: As the precondition of this discussion, as long as two dominant mechanisms coexist, everywhere in a system, dissipations due to the alternating appearance of two corresponding states caused by the interfacial interactions must be governing processes. Using MF-based averaging approaches in this case could lead to the distortion of the real mechanisms,44 hence, losing essential information. Therefore, paying attention to mesoscale structures is the only way if we are still unable to track every individual element.

• No single extremal tendency of total entropy production exists in the mesoregime: Thermodynamics used the total entropy production as its single variational functional, either minimum or maximum, which is suitable in either A- or B-regime, but might not be in the mesoregime,38,39 since this regime is governed jointly by A and B. Turbulence provides a good evidence for this statement. While A reflects laminar flow dominated by the minimum entropy production, B will be purely inertiadominated with the maximum entropy production. However, the A−B regime shows neither minimum nor maximum entropy production due to compromise in competition.44 This might be the reason why turbulence has been so challenging since the 19th century.

Such points might be critical to tackling the difficulties in various disciplines, e.g., to understanding turbulence, to describing or manipulating material structures such as dislocation distributions.45,46 Almost all current global challenges are subject to these points to a certain extent. For instance, current global and regional climate models usually adopt MF-based averaging approaches within a horizontal grid spacing larger than 10 km47 while there are complex (even multilevel) structures in the grid. Using such a grid, how can we define the variational principle it follows? Protein functions are dependent on the change of its dynamic (probably multilevel) structures. Unfortunately, we still frequently use a variational principle defined with minimum free energy. On the one hand, minimum free energy includes two dominant mechanisms minimum internal energy and maximum entropy. How to distinguish the respective effects of these two terms? On the other hand, how to include the influence of its in situ environment or external forces into its variational principle? One more example is the nervous system. How can the sensing information from a huge number of neurons (element) be

• Dissipation extremum is process- and regime-specific: Dissipation is usually believed to be connected to entropy (not the thermodynamic entropy16) production, and usually shows the same extremal tendency as that of the latter. However, dissipation could be related to different 12481

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Industrial & Engineering Chemistry Research integrated by the brain system into a cognitive output? Even for quantum mechanics, if we consider a quantum particle as a system, perhaps the wave−particle duality is correlated with the regime-specific feature, and the uncertainty exists within the mesoregime. As we stated previously,7,38 the uncertainty might originate from some more fundamental certainty. The resolution of all these problems is subject to the breakthrough in understanding the mesoscale behaviors of these systems, for the behaviors at the mesoscales show prominent correlations with those at the element and system scales, and the complexity prevails there. Within the scope of chemistry, we can also raise some rough points for mesoscale modeling in relevant systems, based on the concept of mesoscience:

diffusion equations cannot reproduce the period-doubling instability observed in experiments.31 However, ab initio approaches (even KMC57) are too expensive to reach practical scales, say, tens of micrometers in length and seconds in time. We do need mesoscale models here. Recently we developed a mesoscale model28 for steady states of a model system, but extending it to a dynamic model needs further efforts. • Small systems: Various nanoconstructs are typical small systems in chemistry,58 which might help to realize many great dreams of mankind (e.g., artificial photosynthesis, molecular machines), due to their highly tunable and attractive properties. Although an increasing number of small systems can be handled via ab initio approaches at the element scales with increasing computational capability, multiscale approaches are normally more welcome due to their effectively reduced computational cost,59 and the fluctuation theorem reflects the state-ofthe-art approach at the system scales,58 which we have mentioned above. However, if we unveil the competing mechanisms that govern the mesoscale behavior and shape the mesoscale structures in these systems, through the concept of mesoscience, such systems may be described via mesoscale models. For instance, if we can attribute the action of F0-ATPase60 to the compromise of two competing mechanisms that can be expressed as extremal tendencies, we may describe this behavior via solving the biobjective optimization (or variational) problem. Compared with the common correlative multiscale approaches (where fitting parameters are usually necessary),59 such mesoscale models are expected to give better transferability and more physics.

• Selectivity in complex reactive systems: We analyzed this issue previously,38 suggesting that two kinds of products form alternately with respect to space and time within a specified mesoscale domain, which has been confirmed, to some extent, in the catalytic oxidative cyanation of toluene to benzonitrile.48 The traditional route to predicting selectivity is to analyze the reactive networks. Here detailed (single-event) models49 are at the element scale, usually complicated and massive, and determining the parameters is nontrivial. Therefore, various lumped models have been proposed,50 based on the coarse-graining approach, but improving the predictability, especially the transferability, is still on the way. If we can determine the competing mechanisms corresponding to each of the products (or each kind of products), a mesoscale model may replace the lumped kinetic models. Since such a mesoscale model is based on physical mechanisms, parameter fitting can be greatly reduced (if not totally omitted), and transferability can be expected to be much higher than that for the current lumped kinetic models. • Complex behavior in electrochemistry: Charging Li-ion batteries or supercapacitors can yield complex distributions of ions or electrons, and there exist many relevant reports in the literature.51,52 Since the systems are far from equilibrium, the emergence and evolution of such structures cannot be well described using traditional theories. However, it may be covered within the scope of mesoscience. Although an electric field is applied, the basic processes are still similar to heterogeneous catalysis,28,53 and the role of the electric field might mainly lie in driving diffusion along with chemical driving forces. The processes such as intercalation54 can be treated as reactions here. Following our previous route in analyzing heterogeneous catalysis,53,55 the competing mechanisms here can also be expected to be revealed. Usually, one dominant mechanism leads to one extreme structure (within a limiting regime as well), and corresponds to one kind of processes. Additionally, such dominant mechanisms may be related to different tendencies of entropy production as revealed in our previous work.8,39,44 • Oscillatory behavior in heterogeneous catalysis: Reactive oscillation is well-known in heterogeneous catalysis.56 To accurately describe such behavior, we should go beyond the reaction-diffusion equations where the mean-field approximation has been adopted. For instance, even considering four species, the reaction-

Seemingly, mesoscience offers a new way to overcome the challenges in complex systems. However, mesoscience is still in its infancy, and it is still confronted with many difficulties or limitations. For instance, we have not found a general tool (or sufficient physics) to reveal dominant mechanisms though some hints (correlation with extremal tendencies of energy dissipation rates44) have emerged. Solving multiobjective variational (or just optimization) problems is another apparent obstacle though progress has been made gradually.61 Recently, a novel and promising approach was proposed,62 based on recognizing and analyzing the eigen microstates of statistical ensembles. The concept of ensemble was introduced in 1902 by Gibbs63 and serves as a starting point of statistical physics. An ensemble consists of microstates, which can be obtained from experiments or computer simulations. With microstates described by normalized vectors, the correlations between microstates in an ensemble are defined by their products. Using the eigenvectors of the correlation matrix in this approach, the eigen microstates of a statistical ensemble were introduced,62 and the weight of an eigen microstate in the ensemble is proportional to its eigenvalue. There is a phase transition in a complex system when the weight of one eigen microstate becomes finite,62 and the new phase of the complex system is characterized by the eigen microstate. Near a phase transition point, the dominant eigen microstates exhibit mesoscale structures and their weight factors follow the scaling behaviors with critical exponents. This method has been confirmed in one- and two-dimensional equilibrium Ising models and can be applied in principle to general nonequilibrium complex systems, seemingly providing a powerful 12482

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(18) Veveakis, E.; Regenauer-Lieb, K. Review of Extremum Postulates. Curr. Opin. Chem. Eng. 2015, 7, 40−46. (19) Ross, J.; Corlan, A. D.; Müller, S. C. Proposed Principles of Maximum Local Entropy Production. J. Phys. Chem. B 2012, 116, 7858−7865. (20) Müller, I. A History of Thermodynamics: The Doctrine of Energy and Entropy; Springer: Berlin, 2007. (21) Boltzmann, L. More on Atomism. In Theoretical Physics and Philosophical Problems; Vienna Circle Collection; McGuinness, B., Ed.; Springer: Dordrecht, 1974; Vol. 5. (22) Clausius, R. The Mechanical Theory of Heatwith its Applications to the Steam Engine and to Physical Properties of Bodies; John van Voorst: London, 1865. (23) Glocker, C.; Aeberhard, U. The Geometry of Newton’s Cradle. In Nonsmooth Mechanics and Analysis; Advances in Mechanics and Mathematics; Alart, P., Maisonneuve, O., Rockafellar, R. T., Eds.; Springer: Boston, 2006; Vol. 12. (24) Evans, D. J.; Searles, D. J. The Fluctuation Theorem. Adv. Phys. 2002, 51, 1529−1585. (25) Lesne, A.; Laguës, M. Scale Invariance: From Phase Transitions to Turbulence; Springer: Berlin, 2003. (26) Wang, W.; Lu, B.; Zhang, N.; Shi, Z.; Li, J. A Review of Multiscale CFD for Gas−Solid CFB Modeling. Int. J. Multiphase Flow 2010, 36, 109−118. (27) Breault, R. W. A Review of Gas−Solid Dispersion and Mass Transfer Coefficient Correlations in Circulating Fluidized Beds. Powder Technol. 2006, 163, 9−17. (28) Huang, W. L.; Li, J. Mesoscale Model for Heterogeneous Catalysis Based on the Principle of Compromise in Competition. Chem. Eng. Sci. 2016, 147, 83−90. (29) Wintterlin, J.; Völkening, S.; Janssens, T. V. W.; Zambelli, T.; Ertl, G. Atomic and Macroscopic Reaction Rates of a Surface-Catalyzed Reaction. Science 1997, 278, 1931−1934. (30) Sachs, C.; Hildebrand, M.; Völkening, S.; Ertl, G. Spatiotemporal self-organization in a surface reaction: From the atomic to the mesoscopic scale. Science 2001, 293, 1635−1638. (31) Von Oertzen, A.; Rotermund, H. H.; Mikhailov, A. S.; Ertl, G. Standing Wave Patterns in the CO Oxidation Reaction on a Pt(110) Surface: Experiments and Modeling. J. Phys. Chem. B 2000, 104, 3155− 3178. (32) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations; Wiley: New York, 1977. (33) Li, J.; Tung, Y.; Kwauk, M. Method of Energy Minimization in Multi-Scale Modeling of Particle-Fluid Two-Phase Flow. In Circulating Fluidized Bed Technology II; Basu, P.; Large, J. F., Eds.; Pergamon: London, 1988; pp 89−103. (34) Li, J.; Kwauk, M. Particle-Fluid Two-Phase Flow: The EnergyMinimization Multi-Scale Method; Metallurgical Industry Press: Beijing, 1994. (35) Li, J.; Zhang, Z.; Ge, W.; Sun, Q.; Yuan, J. A Simple Variational Criterion for Turbulent Flow in Pipe. Chem. Eng. Sci. 1999, 54, 1151− 1154. (36) Ge, W.; Chen, F.; Gao, J.; Gao, S.; Huang, J.; Liu, X.; Ren, Y.; Sun, Q.; Wang, L.; Wang, W.; Yang, N.; Zhang, J.; Zhao, H.; Zhou, G.; Li, J. Analytical Multi-Scale Method for Multi-Phase Complex Systems in Process Engineering  Bridging Reductionism and Holism. Chem. Eng. Sci. 2007, 62, 3346−3377. (37) Li, J.; Huang, W.; Edwards, P. P.; Kwauk, M.; Houghton, J. T.; Slocombe, D. On the Universality of Mesoscience: Science of ’the inbetween’, arXiv:1302.5861, 2013. (38) Huang, W.; Li, J.; Edwards, P. P. Mesoscience: Exploring the Common Principle at Mesoscales. National Science Review 2018, 5, 321−326. (39) Li, J.; Ge, W.; Wang, W.; Yang, N.; Huang, W. Focusing on Mesoscales: From the Energy-Minimization Multiscale Model to Mesoscience. Curr. Opin. Chem. Eng. 2016, 13, 10−23. (40) Li, J.; Qian, G.; Wen, L. Gas-Solid Fluidization: A Typical Dissipative Structure. Chem. Eng. Sci. 1996, 51, 667−669.

tool for revealing dominant mechanisms and mesoregimes in mesoscience. We hope mesoscience could be established and forward a powerful tool for tackling complex systems, but more work is still needed as called for recently in a newly shaped international community toward mesoscience.64

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-10-62558318. Fax: +86-1062558065. ORCID

Jinghai Li: 0000-0002-5026-7104 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the International Partnership Program of Chinese Academy of Sciences (Grant No. 122111KYSB20170068) and the National Natural Science Foundation of China (Grant Nos. 21878300 and 91834303) for funding.

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REFERENCES

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Commentary

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DOI: 10.1021/acs.iecr.9b01655 Ind. Eng. Chem. Res. 2019, 58, 12478−12484