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A Statistical Inference of the Principle of Maximum Entropy Production

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A Statistical Inference of the Principle of Maximum Entropy Production Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 2 March 2021 doi:10.20944/preprints202103.0110.v1 Article A statistical inference of the Principle of Maximum Entropy Production Martijn Veening 1,† 1 EntropoMetrics; [email protected] † Current address: Electrastraat 21, 9801 WD Zuidhorn, The Netherlands 1 Abstract: The maximization of entropy S within a closed system is accepted as an inevitability (as 2 the second law of thermodynamics) by statistical inference alone. The Maximum Entropy Production 3 Principle (MEPP) states that not only S maximizes, but S˙ as well: a system will dissipate as fast as 4 possible. There is still no consensus on the general validity of this MEPP, even though it shows 5 remarkable explanatory power (both qualitatively and quantitatively), and has been empirically 6 demonstrated for many domains. In this theoretical paper I provide a generalization of entropy 7 gradients, to show that the MEPP actually follows from the same statistical inference, as that of 8 the 2nd law of thermodynamics. For this generalization I only use the concepts of super-statespaces 9 and microstate-density. These concepts also allow for the abstraction of ’Self Organizing Criticality’ 10 to a bifurcating local difference in this density, and allow for a generalization of the fundamentally 11 unresolved concepts of ’chaos’ and ’order’. 12 Keywords: entropy production maximization, complexity, thermodynamics, nonlinear dynamics, 13 statistical mechanics, autopoiesis, entropy, entropy production, life 14 1. Introduction 15 In the last few decades, several attempts have been made to explain the occurrence 16 and stability of what we call ’living systems’, or ’order out of chaos’. Most notably, 17 the alleged conflict between ’autopoiesis’ and the second law of thermodynamics is still Citation: Veening, M. A statistical 18 unresolved [12]. Despite the impressive progress and multidisciplinary approaches in inference of the Principle of 19 this field, there is still no conclusive consensus on any of these aspects [13]. Maximum Entropy Production. 20 Journal Not Specified 2021, 1, 0. 21 https://doi.org/ This theoretical paper should provide this consensus, not by presenting yet another new 22 theory to describe living systems, but rather by generalizing existing concepts to prove Received: 23 the so-called ’Maximum Entropy Production Principle’, and then infer the occurrence Accepted: 24 and local stability of autopoiesis as a statistical possibility. Published: 25 1.1. Thermodynamic entropy Publisher’s Note: MDPI stays neu- The well-known Boltzmann equation for thermodynamic entropy in a closed system tral with regard to jurisdictional is: claims in published maps and insti- S = −kB ln W (1) tutional affiliations. 26 where W indicates the number of microstates that correspond to a certain macrostate. 27 The well-known Second Law of Thermodynamics tells us that the state of a system will Copyright: © 2021 by the author. 28 tend towards macrostates with a higher W. The trivial example of such a macrostate is a Submitted to Journal Not Specified 29 homogeneous distribution. for possible open access publication 30 It’s irreversibility comes from statistical inference: the state of a system has a higher under the terms and conditions 31 probability to have a macrostate with many microstates, than a macrostate with little of the Creative Commons Attri- 32 microstates. Therefore, over time, it’s W (and S) will increase up to some maximum. bution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Version February 27, 2021 submitted to Journal Not Specified https://www.mdpi.com/journal/notspecified © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 2 March 2021 doi:10.20944/preprints202103.0110.v1 Version February 27, 2021 submitted to Journal Not Specified 2 of 6 33 1.2. Maximization of entropy-production 34 Not only will a system maximize its entropy S over time, it will always do so as fast 35 as possible. This maximization of S˙ is called the ’maximum entropy production principle’ 36 (MEPP) [1][2]. This principle has been empirically validated for many domains, such as 37 the atmosphere [11][6][9], crystallization [5], enzyme reactions [3] and the economy [4]. 38 Intuitively this makes sense, but this principle is still debated, and its ubiquitous validity 39 has not been demonstrated yet [10][7][8]. 40 The materials and methods in this paper will provide for the generalizations neces- 41 sary to finally infer this ubiquitous validity with the same logic as that of the second law 42 of thermodynamics, and then statistically infer the occurrence of ’order’. 43 2. Materials and Methods 44 2.1. Dynamic state-spaces and super-statespaces 45 By convention we consider the state-space of a system having some dimensionality n 46 n 2 N, so we can describe the state of a system as some w 2 R . If we generalize to 47 systems with a dynamic state-space (systems of which the statespace changes over time), 48 then such a statespace itself has a statespace. We provide the following definition: 49 Definition 1. A super-statespace is the statespace of a statespace. 50 The simplest systems then have a zero-dimensional super-statespace with only a 51 single solution, and more complex systems can have super-statespaces with many 52 solutions. 53 An example of a system with a super-statespace with a single solution is the textbook- 54 example of a thermodynamic closed system of some gas that dissipates energy. 55 Examples of systems with a dynamic state-space are complex, non-linear dynamic 56 systems, such as living system, ecosystems, economies, technological systems and 57 financial systems. 58 2.2. Microstate density 59 Entropy is quantified using the number of microstates of some macrostate. To 60 illustrate the entropic gradient (from low to high entropy) one can use a heatmap for 61 the number of microstates. Such a heatmap actually shows the ’microstate density’ of a 62 macrostate. 63 In this sense, the second law of thermodynamics can be illustrated by a trajectory of 64 the state towards an area with a higher microstate-density. Figure1 shows the entropic 65 gradient of a system with a static statespace. It has a zero-dimensional super-statespace 66 (with a single solution, and only a single-dimensional entropic gradient). Figure 1. One-dimensional entropic gradient of a macrostate W 67 A dynamic state-space can contain areas where dimensionality of complexity in- 68 creases (or decreases) over time. This means that in that region of the state-space, there are 69 much more micro-states that correspond to the macrostate: another region with relative 70 high microstate-density. 71 These areas are visualized in Figure2. It clearly shows these ’waves’ of micro-state 72 density of a system with a non-zero-dimensional super-statespace. It shows that sys- 73 tems with non-zero-dimensional super-statespaces can have multidimensional entropic 74 gradients. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 2 March 2021 doi:10.20944/preprints202103.0110.v1 Version February 27, 2021 submitted to Journal Not Specified 3 of 6 Figure 2. Heatmap of micro-state density with 2-dimensional entropic gradients 75 We have now generalized the entropic gradient of systems with a zero-dimensional 76 super-statespace as a special case of multi-dimensional entropic gradients of systems with 77 a super-statespace of any dimensionality (including none). 78 2.3. The inevitability of entropy production maximization 79 We have seen that complex dynamic systems can show multidimensional entropic 80 gradients: either towards higher microstate-density within the same superstate (a static 81 statespace), or towards superstates with a higher dimensional complexity. It then 82 depends on which of these gradients is the steepest (relative to the initial/current state 83 of the system), for the system to follow. 84 So the same statistical inference applies as with the second law of thermodynamics: if 85 the state of a system enters a region with ascending dimensionality (more microstates), 86 the state progresses on this ascent because it has the highest probability to tend towards 87 the state with the most microstates available locally. By definition, this yields the highest 88 amount of entropy production possible. 89 This shows the inevitability and ubiquitous validity of the maximization of entropy 90 production . 91 2.4. Self-organizing criticalities All kinds of external and internal dynamics can alter the state-space, as the state finds the way of maximum entropy production. For example, biochemical structures can collide and form more stable structures, allowing for more complex dynamics. Local maxima of entropy production can emerge and disappear. These can mathematically be described as bifurcations from unstable to stable attractors (e.g. limit cycles) and vice versa, of non-linear entropy production dynamics. These local maxima, where S¨ = 0, are clearly visible in the heatmap in Figure3, which is only a more extreme version of Figure2. These local maxima are the sets of micro-states ~x which satisfy: ¶2S(~x, t) = 0 (2) ¶t2 92 If such a local maximum emerges (and the state of a system is caught in its basin of 93 attraction) we recognize this process as a self-organizing criticality. Eventually such a 94 maximum will disappear again (because of local saturation/exhaustion of entropy pro- 95 duction, roughly speaking), and this implies a collapse of the self-organizing criticality. 96 For organic systems, this is called death. 97 The statespace then decreases in dimensionality again, and resumes its dissipative course, 98 or gets caught in another basis of attraction. 99 In other words, these areas mark the emergence of what we call ’order’, but from this Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 2 March 2021 doi:10.20944/preprints202103.0110.v1 Version February 27, 2021 submitted to Journal Not Specified 4 of 6 100 generalization it is actually an acceleration towards more disorder (albeit local in space 101 and time).
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