4th Tuebingen Summer School in History and Philosophy of Science, July 2015

1. Approximationand IdealizationJohn D. NortonDepartment of History and Philosophy of ScienceCenter for Philosophy of ScienceUniversity of Pittsburgh 4th Tuebingen Summer School in History and Philosophy of Science, July 2015

2. This Lecture 1 2 Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations. Extended example: Thermodynamic and other limits of statistical mechanics. Infinite idealizations are often only limiting property approximations. Stipulate that: “Approximations” are inexact descriptions of a target system. “Idealizations” are novel systems whose properties provide inexact descriptions of a target system. 3 Fruitless debates over reduction and emergence in phase transitions derive from unnoticed differences in the notion of level. reduction theory emergence scale

3. CharacterizingApproximation and Idealization

4. The Proposal Stipulate … Target system (boiling stew at roughly 100oC ) Approximation “The temperature is 100oC.” Inexact description (Proposition) Idealization Another System whose properties are an inexact description of the target system. …and an idealization is more like a model, the more it has properties disanalogous to the target system.

5. A Well-Behaved Idealization Target: Body in free fall Body in free fall in a vacuum dv/dt = g – kv v = gt v(t) = (g/k)(1 – exp(-kt)) = gt - gkt2/2 + gk2t3/6 - … Exact description v = gt Idealization for first moments of fall. Inexact description for the the first moments of fall (t is small). Approximation

6. Approximation only Bacteria grow with generations roughly following an exponential formula. Approximate with n(t) = n(0) exp(kt) fit improves at n grows large. Take limit as n ∞ ?? infinite n(t) infinite n(0) exp(kt) = ?? System of infinitely many bacteria fails to be an idealization.

7. Using infinite Limitsto formidealizations

8. Two ways to take the infinite limit ∞ Idealization Infinite systems may have properties very different from finite systems. The “limit system” of infinitely many components analyzed. Its properties provide inexact descriptions of the target system. ∞ Approximation Systems with infinitely many components are never considered. Consider properties as a function of number n of components. “Properties(n)” “Limit properties” Limn∞Properties(n) provide inexact descriptions of the properties of target system.

9. Limit Property and Limit System Agree Infinite cylinder has area/volume = 2. “Limit system” 4p + 6p , … , , 2 4p + 4p , 4p + 2p “Limit property” 4p/3 + 3p 4p/3 + 2p 4p/3 +p area 4p + 2pa = volume 4p/3 +pa system1, system2, system3, … , limit system agrees with property1, property2, property3, … , limit property Infinite cylinder is an idealization for large capsules.

10. There is no Limit System ? There is no such thing as an “infinitely big sphere.” There is no limit system. Limit property , … , 0 3/1 , 3/2 , 3/3 , area 4pr2 = 3/r = volume 4pr3/3 system1, system2, system3, … , ??? property1, property2, property3, … , limit property Limit property is an approximation for large spheres. There is no idealization.

11. Limit Property and Limit System Disagree Infinite cylinder has area/volume = 2. “Limit system” , , , , … , 3p/4 formula for a=1 formula for a=3 formula for a=4 formula for a=2 “Limit property” area p2a Area formula holds only for large a. = volume 4pa/3 system1, system2, system3, … , limit system DISagrees with property1, property2, property3, … , limit property Infinite cylinder is NOT an idealization for large ellipsoids.

12. When Idealization Succeeds and Fails … 2 2 2 2 limit property Property 1 1 … limit system System N=∞ N=2 N=3 N=1 Limit property and limit system agree Limit property exists, but NO limit system. Limit property and limit system DISagree. Limit system is idealization of large N systems. Limit property is an approximation for large N systems. No idealization.

13. Limits in Statistical Physics

14. Thermodynamic limit as an idealization

15. Two forms of the thermodynamic limit Number of components n ∞ V  ∞ Volume n/V is constant such that Strong. Consider a system of infinitely many components. Weak. Take limit only for properties. “The physical systems to which the thermodynamic formalism applies are idealized to be actually infinite, i.e. to fill Rν (where ν=3 in the usual world). This idealization is necessary because only infinite systems exhibit sharp phase transitions. Much of the thermodynamic formalism is concerned with the study of states of infinite systems.” Ruelle, 2004 Property(n) well-defined limit density volume Le Bellac, et al., 2004. Approximation Idealization

16. Infinite one-dimensional crystal Problem for strong form. Spontaneously excites when disturbance propagates in “from infinity.” then then Determinism, energy conservation fail. This indeterminism is generic in infinite systems. then then

17. Strong Form: Must Prove Determinism Simplest one dimensional system of interacting particles. Clause bars monsters not arising in finite case.

18. Inessential complications…?? “We emphasize that we are not considering the theory of infinite systems for its own sake so much as for the fact that this is the only precise way of removing inessential complications due to boundary effects, etc.,…” Lanford, 1975, p.17 But now extra conditions have to be added by hand to reproduce the essential functions of the boundary conditions.

19. Continuum limit as an approximation

20. Useful for spatially inhomogeneous systems. Continuum limit Number of components n ∞ V fixed Volume nd3 = constant such that Boltzmann’s k  0 Avogadro’s N ∞ Fluctuations obliterated Portion of space occupied by matter is constant. d = component size Continuum limit provides approximation Limit of properties is an inexact description of properties of systems with large n. Idealization fails. No limit state. Stages do not approach continuous matter distribution. See “half tone printing” next.

21. Half-tone printing analogy limit state of gray = everywhere uniformly 50% occupied At all stages of division point in space is black occupied white unoccupied or State at point x = 1/3 y = 2/5 Oscillates indefinitely: black, black, white, white, black, black, white, white, …

22. Boltzmann-Grad limit as an approximation

23. Boltzmann-Grad Limit Useful for deriving the Boltzmann equation (H-theorem). Number of components n ∞ V fixed Volume nd2 = constant such that Portion of space occupied by matter  0 d = component size Limit state of infinitely many point masses of zero mass. Can no longer resolve collisions uniquely. System evolution in time has become indeterministic. Limit properties provide approximation. Idealization fails.

24. Resolving collisions Equations 1 energy conservation 3 momentum conservation 2 direction of perpendicular surface Variables 2 x 3 velocity components for outgoing masses take limit… 6 Lose these for point masses. 6 4

25. Renormalization Group Methods

26. Renormalization Group Methods Renormalization group transformation generated by suppressing degrees of freedom: Best analysis of critical exponents. Zero-field specific heat CH ~ |t|-a … Correlation length x ~ |t|-n … for reduced temperature t=(T-Tc)/Tc N’=bdN clusters of components N components !! Transformations are degenerate if we apply them to systems of infinitely many components N = ∞. such that total partition function is preserved (unitarity): Z’(N’) = Z (N) Hence generate transformations of thermodynamic quantities Total free energy F’ = -kT ln Z = F Free energy per component f’ = F’/N’ = F/bdN = f/bd

27. The Flow space of reduced Hamiltonians Properties of critical exponents recovered by analyzing RNG flow in region of space asymptotic to fixed point =region of finite system Hamiltonians. Analysis employs approximation and not (infinite) idealization. Lines corresponding to systems of infinitely many components (critical points) are added to close topologically regions of the diagram occupied by finite systems.

28. Finite Systems Control Necessity of infinite systems Finite systems control infinite. vs “The existence of a phase transition requires an infinite system. No phase transitions occur in systems with a finite number of degrees of freedom.” Kadanoff, 2000 “We emphasize that we are not considering the theory of infinite systems for its own sake… i.e. we regard infinite systems as approximations to large finite systems rather than the reverse.” Lanford, 1975 Infinite system needed only for a mathematical discontinuity in thermodynamic quantities, which is not observed. …and if it were, it would refute the atomic theory! Properties of finite systems control the analysis.

29. Reduction?Emergence?

30. Phase transitions are… Norton, Butterfield … a success of the reduction of thermodynamics by statistical mechanics. … a clear example of non-reductive emergence. Who is right? BOTH! ..and no one is more right.

31. Different Senses of “Levels” p Molecular-statistical Description.Phase space of canonical positions and momenta.Hamiltonian, canonical distribution, Partition function.Canonical entropy, free energy…. q Few component molecular-statistical level Many component molecular-statistical level Thermodynamic level. State space pressure, volume, temperature, … Internal energy, free energy, entropy, …

32. Where Reduction Succeeds Level of many component, molecular-statistical theory Level of thermodynamic theory deduce Renormalization group flow on space of reduced Hamiltonians. Critical exponents in vicinity of critical points. (Augmented) Nagel-style reduction: deduce surrogate for Lower level theory Higher level theory

33. Where Emergence Happens Few component molecular-statistical level Many component molecular-statistical level A few components • by themselves do not manifest phase transitions • in the mean field of the rest do not manifest the observed phase transition behavior quantitatively. Quantitatively correct results from considering many components and their fluctuations from mean quantities.

34. “More is Different…” P. W. Anderson, Science, 1972. "The constructionist hypothesis [ability to start from fundamental laws and reconstruct the universe] breaks down when confronted with the twin difficulties of scale and complexity. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity, entirely new properties appear...” few atoms--symmetry many atoms—broken symmetry invert do not invert H H H H N N H H Leo Kadanoff "More is the Same…." Journal of Statistical Physics, 137 (December 2009) Phase transitions are “a prime example of Anderson’s thesis.”

35. A conjecture… … but not philosophers suspicious of theories as units. Physiciststend todivide by scale. Philosopherstend todivide by theory. Theory = deductive closure of a few apt propositions. Condensed matter physics deals with systems of many components. Solids, liquids, condensates, … Level = theory Level = processes at same scale Reduction/emergence between systems of few components and many components. Reduction/emergence between self-contained theories of thermodynamic and statistical mechanics. Draw whichever results needed from any applicable theory. Cannot mix results from different theories in one level. (deductive closure) Few-many distinction divides condensed matter physics from atomic and particle physics. Few-many distinction is within one theory. Mean field theory is an approximation, not a level.

36. Conclusion

37. This Talk 1 2 Infinite limit systems can have quite unexpected behaviors and fail to provide idealizations. Extended example: Thermodynamic and other limits of statistical mechanics. Infinite idealizations are often only limiting property approximations. Stipulate that: “Approximations” are inexact descriptions of a target system. “Idealizations” are novel systems whose properties provide inexact descriptions of a target system. 3 Fruitless debates over reduction and emergence in phase transitions derive from unnoticed differences in the notion of level. reduction theory emergence scale

38. The End

39. Appendices

40. Recovering thermodynamicsfrom statistical physics Treated statistically often behaves almost exactly like… Very many small components interacting. Thermodynamic system of continuous substances. Analyses routinely take “limit as the number of components go to infinity.” The question of this talk: how is this limit used? ∞ ?