Emergent proper+es and singular limits: the case of +me- irreversibility. Sergio Chibbaro Institut d Alembert Université Pierre et Marie Curie

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1 Emergent proper+es and singular limits: the case of +me- irreversibility Sergio Chibbaro Institut d Alembert Université Pierre et Marie Curie

2 Introduction: Definition of emergence I J Kim 2000 The whole is more than the sum of its parts (i) Emergence of complex higher-level entities: Systems with a higher level of complexity emerge from the coming together of lower-level entities in new structural configurations. (ii) Emergence of higher-level entities: all properties of higher-level entities arise from the properties and relations of lower-level that characterise their constituents. Some properties of these higher, complex systems are emergent, and the rest merely resultant. (iii)the unpredictability of emergent properties: emergent properties are not predictable from exhaustive information concerning their basic conditions. In contrast resultant properties are predictable from lower-level information. (iv) The unexplainability/irreducibility of emergent properties: Emergent properties, unlike those that are merely resultant, are neither explainable nor reducible in terms of their basal conditions. (v) The causal efficacy of the emergency: Emergent properties have causal powers of their own-novel causal powers irreducible to the causal powers of their basal constituents. Causal powers : beyond our scope

3 Introduction: Reductionism, emergence supervenience R.C Bishop, H Atmanspacher 2006 (i) The description of properties at a particular level of description (including its laws) are both necessary and sufficient conditions to rigorously derive the description of properties at a higher level. This is the strictest possible form of reduction. As mentioned above, it was most popular under the influence of positivist thinking in the mid-20th century. (ii) The description of properties at a particular level of description (including its laws) are necessary but not sufficient conditions to derive the description of properties at a higher level. This version, which we propose calling contextual emergence, indicates that contingent contextual conditions are required in addition to the lower-level description for the rigorous derivation of higher-level properties. (iii) The description of properties at a particular level of description (including its laws) is sufficient but not necessary conditions to derive the description of properties at a higher level. This version includes the idea that a lower-level description are multiple realizations of a particular property at a higher level feature characteristic of supervenience. (iv) The description of properties at a particular level of description (including its laws) are neither necessary nor sufficient conditions to derive the description of properties at a higher level. This represents a form of radical emergence insofar as there are no relevant conditions connecting the two levels whatsoever.

4 Irreversibility Empirical evidence : Time arrow Time-reverse somewhat bizarre for many objects but not for fews Number of objects N capital Issue: Macroscopic irreversibility vs Microscopic reversibility

5 Irreversibility Empirical evidence: Time arrow = second law of thermodynamics Reversible microworld: mechanical laws Thesis: the irreversibility can be seen as an (strong) emergent property in the macroscopic limit A property that does not depend only upon the microscopic dynamics; It requires additional conditions (context): (i) a large number of degrees of freedom (ii) appropriate initial conditions. Historically: philosophical and scientific debate: Radical reversibility: Time arrow is an illusion, the world is reversible (Einstein) Radical irreversibility: Time arrow is a fundamental ingredient, microphysics is wrong (Prigogine)

6 Irreversibility: Boltzmann s vision Microworld: gas molecules follow Boltzmann equation (reversible) for distribution function Kinetic theory Boltzmann equation implies H-Theorem H is related to entropy (with sign -) Naive conclusion: H Theorem proves 2 principle of thermodynamics Naive consequences: macroscopic properties are deduced from microscopic Reductionism

7 Paradoxes against Boltzmann: I Recurrence Recurrence paradoxe by Zermelo: H-Theorem does not fulfil Poincaré s recurrence theorem: Therefore, any growth of the entropy (decrease of H) would be followed by a decrease (increase) Poincaré (Boltzmann) answer:

8 Paradoxes against Boltzmann: II Reversibility Reversibility paradoxe by Loschmidt: The paradox of reversibility may be formulated as follows: Let H decrease from time 0 to time t. At time t, reverse the velocities and start a new evolution with the initial condition thus generated. Then, due to the symmetry of the equations of motion under time reversal, the system should trace back its history and one finds that H increases for a time not shorter than t. Therefore, for any growth of the entropy (decrease of H), also the same decrease (increase) should be possible

9 Paradoxes: Boltzmann s answers Boltzmann intuitive answers: (i) Recurrence: is the remark relevant for our world? example: a cubic centimeter of gas, at room pressure and temperature, (accuracy of positions and 1m/s in speed) time-recurrence years 10 9 m in (ii) The paradox of reversibility arises only if the particles velocities are exactly reversed. Impossible! Preparation macroscopic not microscopic 1 macroscopic state = many microscopic states (i) the recurrence-time for macroscopic (N>>1) objects is enormous (ii) H-theorem is valid since there are many more microscopic configurations which lead to a decrease of H, than configurations leading to its increase. L.B.

10 Irreversibility : solution Deterministic Chaos plays no direct role in irreversibility: However Chaos=sensitivity to initial conditions reversibility paradox irrelevant Exponential growing Chaos of any small perturbation True also for few particles but better for N>>1 Chaos has no role in recurrence T recu τc N Beyond any experimental possibility for N=10,100

11 Irreversibility: singular limit and emergence Why the evolution of f ( x, v,t) implies a monotonic increase of entropy? Boltzmann Hypothesis Stosszahlansatz: two particles are uncorrelated before collisions probabilistic H-Theorem = 2 principle of thermodynamic true statistically In the limit: N T recu H theorem rigorous Irreversibility is an emergent property in the limit of many particles

12 Irreversibility: summary C Cercignani: Molecules= hard spheres diameter σ Number N Limit: σ 0 N but with Nσ 2 const H-theorem holds if initial conditions fulfill molecular chaos If N Molecular chaos is certain (probability=1) L Boltzmann

13 Irreversibility: summary Irreversibility=emergent property of the singular limit from microscopic to macroscopic world Main ingredients: (A) the large number of particles (atoms or molecules), hence the great disparity between microscopic and macroscopic scales; (B) the appropriate initial conditions (molecular chaos). (C) (implicit in B) not all microscopic states evolve in the irreversible fashions predicted by the macroscopic theories, but only their vast majority. In macroscopic systems, containing an enormous number of particles, vast majority actually means practically all microstates, hence irreversibility is a universal fact.

14 Irreversibility: Emergence of different levels of reality Within the same phenomenon levels with different scales Reversibility is not strange for few elements Thermodynamic vision emerge for many molecules a cathedral is different if we look the whole (collective behaviour) or a small detail

15 Conclusions Irony of the history: Boltzmann program apparently reductionist: microscopic explanation of thermodynamic Result of the program: need for additional external ingredients Methodological reductionism useful, if non-dogmatic Birth of a new independent discipline: statistical mechanics Pure logical approach not appropriate for dynamical or not formalised theories or sciences Contemporary examples from exact sciences can be interesting for philosophical or foundation debates Mathematical details unavoidable: importance of singular limits When a singular limit separates two different levels of reality typically novel properties emerge from a non-trivial interaction (downward causation) The form of emergence is strong (aside ontological aspect which remain debatable); levels of reality are independent and deduction is not possible Some relevance for mind or life?

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17 Introduction: Philosophical background and setbacks for emergence Logical positivism, Hempel and Oppenheim (1948) Deductive-nomological model of explanation Explanation as logical deduction Laws of nature Particular conditions Phenomenon Explanans Explanandum Epistemic notion of emergence The issue is whether a given statement (higher level emergent) is deducible from a given set of statements (lower level) condition of derivability Applies also to life as an emergent phenomena Depends on theories and data available Nagel (1961): Notion of inter-theoretic reduction Two cases Non-problematic Problematic: need for bridge principles condition of connectability Criticisms and emergence Advocates of emergence in philosophy of mind Davidson (1970), Bunge (1977), Popper (1977), Sperry (1980, 1986) A renewed philosophical debate Downward causation (e.g. Campbell 1974; Sperry 1986; Kim 1992, 1999) Supervenience (e.g. Kim 1984) Non-reductive physicalism (e.g. Kim 1989, Beckermann 1992) Emergence in biology

18 Introduction: Varieties of emergentism What does it mean to say that the phenomenon P is emergent? Surprise! P generates a feeling of surprise for an observer Unpredictability: P cannot be predicted in advance Never, ever vs. the first time only Exactly vs. to some extent Novelty: P consists in the appearance of novel qualities that did not exist before Non-deducibility: P is non deducible from a set of premises ever vs. given a state of knowledge K Unexplainability: P is not explanable in terms of entities/processes that belong to a lower level of organization Irreducibility: P is not reducible to entities/processes/phenomena that belong to a lower level of organization Ontological vs. epistemic emergence: beyond our scope

19 Irreversibility debate (a) Irreversibility is a basic law of nature and should be taken into account in addition to the Newton s equations (or to the Schrodinger equation in the quantum case). This point of view finds support in the existence of the temporal asymmetry observed in the decay of the neutral K meson [Cronin, 1981]. (b) Irreversibility arises from the fact that real systems are open, i.e. that they interact with their environment. Consequently, they are appropriately described by stochastic processes, which have an irreversible nature. (c) The basic ingredients of irreversibility are deterministic chaos and uncertainties in the initial data. In chaotic systems, uncertainties are amplified very rapidly, so that correlations with the initial state is readily lost, and returning to the initial situations is impossible. (d) The origin of irreversibility resides in the measurement procedure which, at the quantum mechanical level, leads to the collapse of the wave function. (e) The key to irreversibility is the very large number of particles, that are involved.