The Erwin Schrodinger¨ International Boltzmanngasse 9/2 Institute for Mathematical Physics A-1090 Vienna, Austria

ESI NEWS Volume 3, Issue 2, Autumn 2008

Editorial final report of the evaluation was sent to the Austrian Ministry of Science in June 2008. Contents Klaus Schmidt Apart from praising the quality of the Editorial 1 programmes which had taken place at the The 15th anniversary Nobel Prize in Physics 2008 2 ESI during the past five years, he review the ESI in April 2008 panel noted that the Institute had ‘gen- Metastability and Rare Events in was an occasion not tly and wisely’ increased its scope of the Complex Systems 4 only for celebration, programmes in recent years, into areas of but also for an evalu- pure mathematics more remote at present Josef Stefan, Josef Loschmidt and ation of the Institute’s Stigler’s Law 7 from theoretical physics, and into areas of performance over the physics and biology beyond those usually Remarks on Boltzmann’s Philoso- past 5 years and its characterized as mathematical physics. The phy 8 scope for future development. panel felt that this process should be con- The previous evaluation of the ESI ESI News 12 tinued in the same judicious fashion as in in 2003 had been chaired by Nigel recent years. Current and Future Activities of Hitchin (Oxford), who had recruited as The report also contains a number of the ESI 13 co-evaluators Robbert Dijkgraaf (Ams- terdam), Jurgen¨ Jost (Leipzig), Nicolai recommendations for the development of Schrodinger¨ Lectures 2008/2009 15 Reshetikhin (Berkeley) and Vincent Ri- the ESI over the next years. These pro- posals and their financial implications are Senior Research Fellows’ Lecture vasseau (Orsay). currently under further discussion with the Courses 2008/2009 15 The evaluation in 2008 followed essen- tially the same pattern: Peter Goddard (IAS University of Vienna and the Austrian Min- Workshop on Mathematics at the Princeton) agreed to chair the evaluation istry of Science. Turn of the 20th Century 15 and chose for his panel of co-evaluators Jean-Michel Bismut (Orsay), Robbert Di- On behalf of the Erwin Schrodinger¨ In- jkgraaf (Amsterdam), Felix Otto (Bonn) stitute I would like to send seasons greet- and Scott Sheffield (Courant Institute). Af- ings and my best wishes for a happy and ter a site visit by the panel in April 2008 the peaceful year 2009.

EVALUATION OF THE ESI, APRIL 2008

The scientific directors of the ESI, Joachim Schwermer and Jakob Yngvason, together with the evaluators Peter Goddard (IAS Princeton, chair), Scott Sheffield (Courant Institute), Jean-Michel Bismut (Orsay) and Felix Otto (Bonn) – from left to right. 2 Volume 3, Issue 2, Autumn 2008 ESI NEWS Nobel Prize in Physics Till the middle of the last century, symme- The main inspiration for Nambu came try principles were thought to express the from the BCS theory of superconductivity 2008: Symmetry Violation basic simplicity of nature. The discovery [2]. In certain metals, the electron-phonon in Subatomic Physics that some of the “sacred” symmetries such interaction can lead to the formation of as parity (P), time reversal or charge con- Cooper pairs (two electrons with opposite Gerhard Ecker and Walter Grimus jugation (C) were only approximate came momenta and spins) that condense in the therefore as a surprise. In his Nobel Lec- ground state. The ground state is there- tures of 1979 Weinberg expressed this feel- fore charged implying the spontaneous vi- The No- ing as a question: “Is nature only approxi- olation of gauge invariance. In an impor- bel Prize in mately simple?” By now there is a general tant paper [3], Nambu demonstrated that Physics 2008 consensus that also approximate symme- gauge invariance, although not manifest in was awarded tries may reveal fundamental properties of the BCS theory, is nevertheless fully main- to Yoichiro nature. For instance, CP violation is essen- tained. Nambu (Enrico tial for the baryon asymmetry of the uni- That was the starting point for Nambu Fermi Institute, verse and thus for our existence [1]. and Jona-Lasinio to investigate the struc- Chicago, USA) for “the discovery of the Symmetries in particle physics can be clas- ture of the ground state in relativistic mechanism of spontaneous broken symme- sified in four groups: quantum field theories relevant for parti- try in subatomic physics” and to Makoto cle physics. The Lagrangian of the Nambu– Kobayashi (KEK, Tsukuba, Japan) and 1. Exact symmetries leave the field Jona-Lasinio (NJL) model [4] Toshihide Maskawa (Yukawa Institute for equations and the ground state µ Theoretical Physics, Kyoto, Japan) for “the of the theory invariant. Examples LNJL = ψγ i∂µψ+ are the Poincare´ transformations discovery of the origin of the broken sym-   metry which predicts the existence of at (Lorentz transformations and space- g0 ψψψψ − ψγ5ψψγ5ψ time translations) and the gauge least three families of quarks in nature.” describes the self-interaction of a single nu- symmetries of electromagnetic and cleon field ψ and is modeled after the BCS strong interactions. Symmetries in physics Lagrangian. Because of the absence of a The modern view of symmetries as groups 2. Spontaneously broken symmetries mass term mψψψ the Lagrangian and the of transformations that leave the equations still leave the field equations invari- field equations are invariant with respect of motion invariant emerges in the 19th ant, but not the ground state. Nambu to two independent phase transformations century. With the advent of quantum me- was the first to realize the impor- (chiral symmetry) chanics in the 1920s, symmetries were tance of this concept in relativistic ψ(x) → eiαψ(x), ψ(x) → eiβγ5 ψ(x) . found to explain degeneracies in spectra quantum field theories. Examples are and to give rise to relations between mea- the chiral symmetry of the strong in- Depending on the size of the coupling surable quantities such as cross sections teractions (in the limit of vanishing constant g , the ground state of the the- and decay rates. In addition to the classical 0 quark masses) and the electroweak ory may exhibit spontaneous breakdown of space-time symmetries (Galilei, Poincare),´ gauge symmetry. the chiral symmetry, generating both a fi- a new type of symmetries occurs in quan- nite mass for the initially massless nucleon tum field theories: the so-called “internal” 3. Approximate symmetries do not and a massless collective excitation (pion). symmetries do not affect space and time even leave the field equations in- The modern version of the NJL model but they transform the fields of the theory. variant, but traces of the symme- is Quantum Chromodynamics (QCD), the The gauge symmetries of the fundamental tries are still visible in measurable gauge theory of quarks and gluons for the interactions are prominent examples. quantities. Symmetries violated only strong interaction. Chiral symmetry holds by the weak interactions (P, CP, Symmetries play an important role in sub- for massless quarks, which is a good ap- strangeness, . . . ) are of this type. atomic physics: proximation for the two lightest quarks Kobayashi and Maskawa received u, d. Spontaneous breaking of chiral sym- i. Unlike gravitational and electromag- the Nobel Prize for their explanation metry provides the only plausible expla- netic forces, nuclear forces are re- of CP violation in the SM. nation why pions are by far the lightest mote from our everyday experience. hadrons. From the study of spectra and other 4. Anomalous symmetries leave the observables, physicists extract clues classical field equations invariant, In the NJL model, most of the nucleon for underlying symmetries to be but are violated by quantum effects. mass arises from spontaneous chiral sym- implemented in the corresponding metry breaking. According to our present quantum field theories. Spontaneous symmetry breaking understanding, the small u and d masses The notion of spontaneous symmetry contribute actually only about 5% to the ii. Theoretical physics often proceeds breaking originated in solid state physics. nucleon mass. In other words, 95% of our by analogies and extrapolations: ex- For example, the Hamiltonian describing a mass and of the visible universe are due to tending the gauge symmetry of the ferromagnet is rotationally invariant in the the strong quark-gluon interaction in QCD. electromagnetic interaction to the absence of an external magnetic field, yet Only the small remainder requires a differ- weak and strong interactions led the ground state exhibits spontaneous mag- ent mechanism for mass generation. to the modern theory of the fun- netization along some arbitrary direction: Inspired by the work of Nambu and Jona- damental interactions, the Standard rotational symmetry is spontaneously bro- Lasinio, Goldstone demonstrated [5] that Model (SM) of particle physics. ken in the ferromagnet. spontaneously broken symmetries lead to

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 20083 massless excitations in general. In the case V–A (vector minus axial vector current) Hamiltonian density is given by of so-called global symmetries (e.g., chi- theory of weak interactions in which both ral symmetries), these excitations would be P and C are maximally violated. Hcc = massless particles (pions for mu = md = g µ +
In the V–A theory, CP was conserved in √ u¯γ (1 − γ5)V d W + H.c. , 0). As shown a few years later by Brout µ a natural way. Therefore, the discovery 2 2 and Englert [6] and by Higgs [7], the situ- + of the decay K → π+π− in 1964 at where W is the W-boson field, g is the ation is different for gauge symmetries (lo- L the Brookhaven National Laboratory [8], SU(2) gauge coupling constant and u and cal symmetries): the Nambu-Goldstone ex- a manifestation of CP violation, was com- d are the column vectors containing the citations are not physical particles but they pletely unanticipated and provided a new quark fields with charge 2/3 and −1/3, re- render some of the initially massless gauge puzzlement which did not diminish with spectively. In Hcc, the quark fields are as- vector fields massive. This is the relativis- the advent of the SM of electroweak inter- sumed to be mass eigenfields. The matrix tic analogue of the Meissner effect in su- actions, in which the V–A theory found a V , nowadays called Cabibbo–Kobayashi– perconductivity. In particle physics, it is home as a certain low-energy limit. Maskawa or CKM matrix, appears as a known as the Higgs effect of spontaneous consequence of the diagonalization of the gauge symmetry breaking. The SM was formulated with leptons and complex mass matrices for the quarks of quarks. To account for the hadrons known both charges. It is a general n × n uni- Based on the foundations laid by Nambu, at that time, three quarks u, d and s were spontaneous symmetry breaking is an es- tary matrix, where n is the number of quark sufficient. The mixing between d and s— families. It is an easy mathematical task to tablished phenomenon in the SM. Sponta- the famous Cabibbo angle—could straight- neous breaking of the electroweak gauge show that, if one absorbs 2n − 1 phases forwardly be transferred from pre-quark into the fields u and d, then the number of symmetry is responsible for the masses of models [9] to the SM. However, apart W and Z bosons, the carriers of the weak phases left over in V is (n − 1)(n − 2)/2. from the CP problem, another unexplained These phases are physical and their number interactions, and for the masses of quarks experimental fact remained, namely the and charged leptons. However, the specific is zero for one or two quark families. How- absence of flavour-changing neutral cur- ever, for n = 3 there is exactly one phase implementation of this mechanism remains rents at the tree level, manifesting itself to be uncovered. It was one of the main which produces CP violation; this is the in the severe suppression of decays like important observation made by Kobayashi motivations for building the collider LHC + + + − K → π e e . This problem was solved and Maskawa. The discovery of the bottom at CERN to unravel this mechanism. The in 1970, when Glashow, Iliopoulos and Higgs model of spontaneous gauge sym- quark in 1977 as a member of a third quark Maiani proposed the existence of a fourth family promoted the Kobayashi–Maskawa metry breaking describes only one of the quark, the charm quark c, which provided a possible scenarios. mechanism to a respectable and realistic very simple mechanism to remove flavour- candidate for the description of CP viola- changing neutral currents, the so-called The original work of Nambu was con- tion. The existence of the second member GIM mechanism [10]. cerned with the strong interaction. His dis- of the third quark family, the top quark, was coveries paved the way for the formula- This was the situation when Kobayashi and established much later in 1995. tion of an effective quantum field theory Maskawa, at that time both at Kyoto Uni- Until the year 2000, our knowledge about of hadrons at low energies (E  1 GeV) versity, wrote their famous paper [11]. Al- CP violation came exclusively from the where the usual perturbative treatment of though the c quark was discovered only in K0K¯ 0 system. Only after 2000, with the QCD cannot be applied because of the 1974, they took the “quartet scheme”, i.e. advent of the results of the Belle ex- permanent binding of quarks and gluons the existence of four quarks, for granted. periment in Tsukuba, Japan, and of the (confinement). In the words of the No- The bulk of the paper consists of the proof BABAR experiment in Stanford, U.S.A., bel Committee, this effective field theory that for all possible assignments of the plenty of CP-violating observables were called Chiral Perturbation Theory has be- four left-handed and the four right-handed measured in B-meson systems, and the come the “. . . standard tool to compute quark fields to multiplets of the SM gauge Kobayashi–Maskawa theory of CP viola- strong interaction processes in this energy group SU(2) × U(1), CP violation was ei- tion was confirmed in a spectacular way. range . . . ” The Particle Physics group at the ther absent or, if present, there was a con- One could ask oneself why CP violation University of Vienna has provided and ap- tradiction to experiment. Thus they con- is so elusive. This is intimately connected plied some of those tools during the past 20 cluded that “no realistic models of CP vi- with the small mixing angles in V . Actu- years. olation exist in the quartet scheme without ally, using the fact that different columns introducing new fields.” One of their sug- of V are orthogonal, one can write down Explicit breaking of the CP symmetry gestions for new fields was a second Higgs relations like doublet, an extension of the SM that is still In the second half of the fifties, inconsis- ∗ ∗ ∗ investigated today. Nearly at the very end VudVub + VcdVcb + VtdVtb = 0. tencies in the interpretation of weak-decay of the paper they suggested the introduc- data became a pressing problem. In 1956, The three terms in this sum form a trian- tion of a third family of quarks, a coura- in a beautiful theoretical paper, Lee and gle in the complex plane. All such “uni- geous venture at a time when not even the Yang proposed P violation in weak inter- tarity triangles” that can be obtained from fourth quark had been discovered, which actions as a resolution to this problem and V have the same area. This area is a mea- eventually earned them the Nobel Prize. suggested to test this hypothesis in β-decay sure for the strength of CP-violating ef- of polarized nuclei or particles. Soon there- Why is the number of quark families con- fects. The small quark-mixing angles imply after, P violation was confirmed in the de- nected with CP violation? The point is that a small area of approximately 1.5 × 10−5, cay of 60Co and, nearly simultaneously, in CP violation is caused by complex cou- of the order of the sixth power of the sine muon decay. The successful conclusion of pling constants in the interaction Hamilto- of the Cabibbo angle [12]. Therefore, CP- this fundamental issue was provided by the nian. The weak charged-current interaction violating effects are suppressed despite a http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 4 Volume 3, Issue 2, Autumn 2008 ESI NEWS Gerhard Ecker and Walter Grimus are at [6] F. Englert and R. Brout, Phys. Rev. large CP-violating phase. In that sense, CP the Faculty of Physics, University of Vi- Lett. 13 (1964) 321. is an approximate symmetry even of the enna. weak interactions. In other words, if CP vi- [7] P.W. Higgs, Phys. Rev. Lett. 13 olation in a process is large, then the pro- (1964) 508. cess is rare; if a process is not rare, then CP References [8] J.H. Christenson, J.W. Cronin, V.L. violation is suppressed in this process. [1] A.D. Sakharov, Pisma Zh. Eksp. Teor. Fitch and R. Turlay, Phys. Rev. Lett. The Kobayashi–Maskawa theory of CP Fiz. 5 (1967) 32 [JETP Letters 5 13 (1964) 138. violation is a stunning success story and (1967) 24]. will most probably experience further con- [9] M. Gell-Mann and M. Levy,´ Nuovo [2] J. Bardeen, L.N. Cooper and J.R. firmation by future measurements. On the Cim. 16 (1960) 705; Schrieffer, Phys. Rev. 106 (1957) other hand, this mechanism of CP viola- N. Cabibbo, Phys. Rev. Lett. 10 162; ibid. 108 (1957) 1175. tion will have its limitations just as the (1963) 531. SM with which it is inseparably connected. [3] Y. Nambu, Phys. Rev. 117 (1960) One limitation is already known: in order 648. [10] S.L. Glashow, J. Iliopoulos and L. to explain the baryon asymmetry of the Maiani, Phys. Rev. D 2 (1970) 1285. [4] Y. Nambu and G. Jona-Lasinio, Phys. universe, the Kobayashi–Maskawa mecha- Rev. 122 (1961) 345; ibid. 124 (1961) [11] M. Kobayashi and T. Maskawa, Prog. nism is not sufficient. Such an explanation 246. Theor. Phys. 49 (1973) 652. requires an extension of the SM with new sources of CP violation. [5] J. Goldstone, Nuovo Cim. 19 (1961) [12] C. Jarlskog, Phys. Rev. Lett. 55 154. (1985) 1039.

Metastability and Rare acterized by widely different time scales. ulator who attempts to study the freezing Consider, for instance, the freezing of su- of water with molecular dynamics simula- Events in Complex Systems percooled water. The fastest motions that tion. The number of steps required to ob- Christoph Dellago need to be taken into account in a molec- serve one single event simply lies far be- ular dynamics simulation of this process yond the possibilities of current (and fu- are bond vibrations and molecular libra- ture) computer systems. Similar compli- Since its invention in tions with periods of tens of femtoseconds cations arise in all systems in which en- the 1950s, molecular as well as molecular reorientation taking ergy barriers or entropic bottlenecks hin- dynamics simulation place on the picosecond scale. To faithfully der the motion of the system and partition has developed into a reproduce these rapid motions, time steps phase space into metastable basins. The powerful and versatile in the femtosecond range are necessary. dynamics of such systems is then charac- tool in the physical sci- In contrast, the time scale for the crystal- terized by infrequent but rapid transitions ences. Today, running lization event is many orders of magnitude between long-lived states: we speak about on modern comput- larger. It has been known for a long time rare events. ers, molecular dynamics simulations are that a sample of water, carefully cooled An important concept to deal with rare used to study the structure and dynamics below the freezing point, can remain in events is transition state theory, developed of complex systems consisting of up to a the metastable liquid state for hours if not in the context of chemical kinetics by few million atoms of interest in physics, days, and only an external perturbation ini- Eyring, Polanyi and Wigner [1, 2, 3] in the materials science, chemistry and biology. tiates the freezing process that turns the liq- 1930s, long before the advent of fast elec- In this method, the motion of individual uid into the thermodynamically stable crys- tronic computers. The central idea of tran- atoms is followed in time by integrating talline phase. sition state theory is that during transitions the classical equations of motion for small The reason for this behavior is that between stable states separated by a high time steps with forces obtained from em- freezing proceeds through the formation of energy barrier, the system passes through pirical interaction potentials or ab initio a critical nucleus involving the creation of a saddle point in the potential energy sur- by solving the electronic structure prob- an interface between the crystallite and the face. This so called transition state pro- lem. By iterating this basic step, the time metastable liquid phase. Due to the free en- vides the lowest energy passageway con- evolution can be determined, in principle, ergetic cost associated with the the inter- necting the stable states located at poten- for arbitrarily long time. But in spite of face, the process is uphill initially leading tial energy minima. From the properties the rapid growth in raw computing speed to a free energy barrier that opposes rapid of the transition state one can then calcu- and impressive algorithmic advances, nu- solidification. Only once the crystalline nu- late reaction rates and glean information on merous processes occurring in nature and cleus grows beyond the critical size due to the microscopic transition mechanism. In technology are still beyond the reach of a rare statistical fluctuation, does the vol- complex condensed phase systems, how- current technology. ume term in the free energy prevail and ever, this simple picture on which transi- One of the factors that limits the prac- the phase transition occurs. Naturally, such tion state theory is based breaks down. In a tical applicability of molecular dynamics long nucleation times, which can exceed condensed phase environment, stable states simulation is that many processes, such as the basic timescale of molecular motion typically cease to be associated with indi- the folding of a protein or the transport of a by many orders of magnitude, present an vidual minima in the potential energy sur- dopant through a semiconductor, are char- enormous challenge for the computer sim- face. Rather, they can encompass a large

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 20085 number of local minima and saddle points discussed during the program. For a more Finding rare transition pathways (the supercooled liquid from above, for in- thorough discussion of new developments In a molecular dynamics simulation stance, is certainly a long-lived metastable in this area, I refer the reader to a recent one specifies initial conditions and then state but does not correspond to a particu- review article [4]. lets the system evolve freely according to lar potential energy minimum). Similarly, the underlying equations of motion. The transition states can no longer be pictured Exploring configuration space system then explores configuration space as individual saddle points. This situation without external guidance and often things is depicted in Fig.1 which represents an Configuration space is a big place. To happen serendipitously that could not have artist’s view of the potential energy surface explore all of its important regions is a been easily anticipated. In the metadynam- in a complex high-dimensional system. challenging task, particularly if sampling ics method, one tries to accelerate this natu- is complicated by the presence of widely ral time evolution by preventing the system disparate time scales. One strategy to re- from returning to configuration space re- duce the complexity of the problem is to gions that have been sampled before. As in introduce a small set of collective variables conventional molecular dynamics, the dy- which are presumed to include all degrees namics starts from given initial conditions of freedom that are important for the pro- and the state the system evolves to is not cess of interest and cannot be replaced by specified in advance: both approaches are random noise. The free energy landscape, single-ended methods. determined as a function of these collec- In some situations, however, both the tive variables, then provides the informa- initial and the final state of a certain pro- tion necessary to locate stable regions and cess are known, but not the mechanism identify possible routes for transitions be- that transports the system from one to the Figure 1: Low-dimensional caricature of a tween them. Of course, such a simplified other. Think, for instance, of the crystal- high-dimensional potential energy surface picture obtained by integrating out most lization of supercooled water considered with multiple stationary point. Two deep degrees of freedom is accurate only if the above. Both the supercooled liquid as well stable states, containing several local min- right collective variables have been chosen, as the ice crystal are well known and char- ima and saddle points, are separated by a an often exceedingly difficult problem. acterized, the detailed freezing mechanism rugged free energy barrier. While several computational methods on an atomistic scale, however, is still un- such as umbrella sampling [5] or thermo- certain. Similarly, in many structural phase In studying processes occurring on dynamic integration [6] for the computa- transitions occurring in solids, one exactly such energy landscapes a number of non- tion of free energies as a function of one knows the stable and metastable phases, trivial questions arise that go beyond a or two variables have been available for but has only vague ideas about the atom- mere time scale problem in the molecu- quite some time, only recently it has be- istic transition mechanism. In the past cou- lar dynamics simulation. How, for instance, come possible to compute free energies ple of years, some advances have been can the long-lived states be identified in in higher dimensions. The turning point made in the development of double-ended terms of the microscopic phase space co- came with the invention of the metady- path based methods designed to address ordinates? Which variables can be used to namics method by Laio and Parrinello in this problem of rare transitions between describe the transition and serve as a re- 2002 [7]. The simple and effective ba- known long-lived states. action coordinate? What are the transition sic idea of this method is to run molec- One approach that has been applied to a states? How often do transitions occur? In ular dynamics on a biased energy land- large variety of rare event problems is tran- the past couple of years several computa- scape. The bias is built up on the fly in a sition path sampling, mainly developed by tional tools to simulate processes involving way to force the system to explore phase Dellago, Bolhuis and Chandler [8, 9]. The rare but important events and address such space regions it has not visited before. conceptual foundation of transition path questions have been put forward. These This is achieved by leaving behind repul- sampling is the statistical definition of the new approaches and related issues were sive Gaussian potentials acting on the pre- set of all dynamical pathways that start and the focus of the ESI-program on Metasta- selected collective variables. These poten- end in given regions of configuration space. bility and Rare Events in Complex Sys- tials effectively mark the already visited This set of pathways, the transition path en- tems, February-April 2008, organized by regions and drive the system into unex- semble, is then sampled with a Metropo- C. Dellago, P. G. Bolhuis, and E. Vanden- plored territory. As a result, one obtains lis Monte Carlo algorithm. The basic step Eijnden and attended by most of the re- possible transitions routes to other stable of this procedure consists in generating a searchers active in this field. One of the states and, as a bonus, the free energy as new path from an old one, for instance by main goals of this workshop was to bring a function of the collective variables. The shooting off the new trajectory from a point together people working on rare events in bias, however, perturbs the natural dynam- on the old one with slightly perturbed mo- different areas ranging from physics and ics of the system such that dynamical quan- menta. Then, the new path is accepted or chemistry to materials science and molec- tities, e.g., transition rate constants, can- rejected according to a criterion that satis- ular biology. In particular, the hope was not be computed with metadynamics. To fies detailed balance, thus guaranteeing that to involve mathematicians and to establish date, metadynamics has been successfully the desired path ensemble is sampled. Iter- closer links between researchers engaged applied to study numerous processes occur- ation of this step generates a biased random in methods development and those working ring in condensed phases including struc- walk in the space of trajectories, in which on large scale applications. In the follow- tural phase transitions in solids, chemi- pathways are visited according to their sta- ing, I will briefly (and incompletely) sur- cal reactions, and biomolecular isomeriza- tistical weight in the transition path ensem- vey a few of the key ideas and concepts tions. ble. http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 6 Volume 3, Issue 2, Autumn 2008 ESI NEWS computer can provide valuable insight and one important challenge will be to com- While the transition path sampling pro- assist one’s imagination, but too often this bine the complementary strengths of these cedure contains a random element, indi- exercise is a sobering experience as the im- methods in order to efficiently cross the vidual pathways are fully dynamical tra- portant degrees of freedom capturing the vast time-scale gap that lies between the jectories governed by the deterministic or essential physics of the process remain elu- macroscopic and the microscopic world. stochastic equations of motion of the sys- sive. Here, statistical analysis tools devel- Hopefully, the ESI-program on Metasta- tem. Therefore, pathways harvested with oped recently help to make progress. bility and Rare Events in Complex Systems transition path sampling can be analyzed The central concept, on which most of has helped to create new momentum and to yield the mechanism as well as the ki- these algorithms build, is the committor, interactions for concerted efforts in this netics of the transition. Since only reac- an idea going back at least to Onsager, direction. tive pathways are considered and typically who used it to analyze the dissociation of rare barrier crossing events occur rapidly ions in solution [12]. This quantity, defined Christoph Dellago is at the Faculty of once they happen, no computing power is for a particular configuration, measures the Physics, University of Vienna wasted to follow the time evolution dur- probability that dynamical trajectories ini- ing the uneventful permanence in the sta- tiated from that configuration relax into one ble states. Current research efforts are di- References of the stable states rather than the other. [1] H. Eyring, J. Chem. Phys. 3, 107 rected towards the development of more ef- The committor is the ideal reaction coor- (1935). ficient transition path sampling algorithms dinate in the sense that it quantifies how far for the computation of rate constants and a reaction has proceeded and what is likely [2] H. Eyring and M. Polanyi, Z. Phys. the ergodic sampling of pathways in sys- to happen next. While configurations near Chem. B 12, 279 (1931). tems with multiple transition routes. To the stable states typically have committors date, the transition path sampling method- [3] E. Wigner, Trans. Faraday Soc. 34, 29 close to 0 or 1, configurations with a com- ology has been used to study to a wide (1938). mittor of 1/2 located on top of the (un- variety of processes in physics, materials known) free energy barrier can be viewed Adv. science, chemistry, and biology. Applica- [4] C. Dellago and P. G. Bolhuis, as transition states. These are states from Poly. Sci. tions range from first order phase transi- , in print (2008). which both stable states are equally acces- tions to chemical reactions in solution and sible. With this statistical generalization of [5] G. M. Torrie and J. P. Valleau, J. biomolecular conformational changes. The the concept of the transition state, an en- Comp. Phys. 23, 187 (1977). transition path sampling method, including semble of transition states can be deter- applications, is reviewed in [4]. [6] J. Kirkwood, J. Chem. Phys. 3, 300 mined from which information on the tran- (1935). Another recent approach to study tran- sition mechanism can be inferred. sitions between known stable states dis- While the committor is, in a sense, the [7] A. Laio and M. Parrinello, Proc. Natl. cussed during the program at the ESI is perfect reaction coordinate, it is very un- Acad. Sci. USA 99, 12562 (2002). the string method developed by E, Ren, specific and it does not directly lead to a [8] C. Dellago, P. G. Bolhuis, F. S. and Vanden-Eijnden [10]. In this method, true physical understanding of the transi- Csajka, and D. Chandler, J. Chem. suitable for systems evolving according to tion mechanism. The committor is, how- Phys. 108, 1964 (1998). stochastic dynamics and rooted in transi- ever, very useful for testing the quality of a tion path theory [11], a string with end- postulated reaction coordinate [13]. While [9] C. Dellago, P. G. Bolhuis, and P. L. points anchored in the stable states is a good reaction coordinate parametrizes Geissler, Adv. Chem. Phys. 123 1 adapted iteratively according to an ap- the committor, no such relation exists for (2002). propriate protocol until convergence is a poor reaction coordinate. This idea is reached. The converged string can be exploited in two computational methods [10] W. E, W. Ren, and E. Vanden- viewed as a typical representative of high for the automatic identification of reaction Eijnden, Phys. Rev. B 66, 052301 likelihood trajectories connecting the sta- coordinates, based on genetic neural net- (2002). ble states, from which rates and mechanism works [14] and likelihood maximization [11] W. E and E. Vanden-Eijnden, J. Stat. can be extracted. [15]. There is no doubt that these and simi- Phys. 123, 503 (2006). lar analysis methods will play a major role Identifying mechanisms in future studies of rare event processes. [12] L. Onsager, Phys. Rev. 54, 554 Transition path sampling and other (1938). methods for the simulation of rare events Outlook typically yield many realizations of the The computational techniques and the- [13] P. L. Geissler, C. Dellago, and D. transition event. Further analysis is most oretical concepts briefly presented here Chandler, J. Phys. Chem. B 103, 3706 often required to identify the exact tran- and discussed at length during the ESI- (1999). sition mechanism in terms of a reaction program have been available only for a [14] A. Ma and A. R. Dinner, J. Phys. coordinate, i.e., a variable that quantifies couple of years. Nevertheless, they have Chem. B 109, 6769 (2005). the progress of the transition. Sometimes, already contributed significantly to our un- watching the motion of individual atoms derstanding of many processes in complex [15] B. Peters and B. L. Trout, J. Chem. with a molecular viewing program on a condensed matter systems. In the future, Phys. 125, 054108 (2006).

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 20087

Josef Stefan, Josef Stefan recog- perature difference, especially at high tem- peratures. Although various models were Loschmidt and Stigler’s nized Loschmidt’s brilliance and sub- proposed, and some were widely incorpo- Law sequently hired rated, Stefan felt that they were not partic- ularly good. So he compiled the data from John Crepeau him as a re- searcher at the In- a number of different experiments and pos- 4 stitute. While at tulated his T model. This model was due Josef Stefan1 and Josef there, Loschmidt in large part to the experiments by Tyn- Loschmidt2 were both became interested dall, who performed experiments on heated affiliated with the Insti- in James Clerk platinum wire. There were flaws in Tyn- tute of Physics at the Maxwell’s kinetic theory of gases. He fig- dall’s data, which by happy circumstance University of Vienna in ured that one way to calculate the size cancelled out, allowing Stefan to postu- the latter half of the of a molecule would be to measure the late the correct dependence of the energy 1800s; Stefan as Direc- density of liquefied gas, and assume that on the temperature for this mode of heat tor and Loschmidt as a the molecules in liquid form were tightly transfer. Boltzmann subsequently derived researcher. Both came from poor rural packed together, occupying a volume only the relation from thermodynamic princi- 4 families in the Austro-Hungarian empire, slightly larger than the volume occupied by ples, and the T law is known as the Stefan- 11 both made seminal contributions to sci- the molecules themselves. From this basis, Boltzmann law . ence, but both have been recognized differ- he used the kinetic gas equations developed ently within the scientific community. Ste- by Maxwell and Rudolf Clausius to suc- As a member of the Austrian Academy fan’s name3 is associated with constants, cessfully calculate the size of a molecule. of Sciences, Stefan was privy to the latest physical laws and dimensionless numbers, Maxwell himself used Loschmidt’s meth- scientific data. At one meeting, he heard while Loschmidt4 has remained virtually ods to calculate the size of gas molecules a talk given by Karl Weyprecht, a leader anonymous, his contributions usurped by in order to determine the diffusion coeffi- of the Austro-Hungarian Polar Expedition, others. cients of various gases. who presented data on the rate of ice growth in the Polar Sea. Based on this in- Stigler’s Law of Eponymy5 states, “No Maxwell stated, “Loschmidt, in 1865, formation, as well as the data presented scientific discovery is named after its orig- made the first estimate of the diameter of a by similar expeditions, Stefan became in- inal discoverer.” This sociological observa- molecule”, and then used Loschmidt’s val- terested in modeling solid-liquid phase tion shows that throughout the history of ues to calculate that, “in a cubic centime- change. The difficulties of this problem science, many who have made outstanding ter of gas at standard pressure and temper- lay in the time dependent, moving bound- contributions are often not rightfully recog- ature there are about nineteen million mil- aries, which locations were not known a nized. Among the great contributions made lion million molecules”.8 priori. By performing an energy balance by Josef Loschmidt, two stand out as ap- Ludwig Boltz- across the interface between the solid and plications of Stigler’s Law. After working mann, a student liquid phases, Stefan was able to model in the chemical industry for a number of 12 both of Stefan and the growth rate of the solid . This mov- years, he longed to perform research, but Loschmidt, first pro- ing boundary problem is now called the was unable to gain a suitable position. So, posed that the num- Stefan problem, and the associated dimen- he became a schoolteacher with the added ber of particles in a sionless variable is called the Stefan num- perk that the school provide a small lab- cubic centimeter be ber. However, Stefan was not aware that the oratory in which to do work in his spare called Loschmidt’s same problem had been already indepen- time. Under these difficult circumstances, number,9 stating this dently investigated by Lame´ and Clapeyron Loschmidt published, at his own expense, was Loschmidt’s as well as Neumann. So Stefan’s name lives a booklet entitled Chemische Studien I6, 7 greatest accomplish- on for work that he did not do first, or was (no part II ever appeared), where he dia- ment. Despite Maxwell’s and Boltzmann’s based on flawed data. grammed single, double and triple chemi- explicit acknowledgement that Loschmidt cal bonds, as well as a novel way to con- Despite the contributions of both Ste- provided the means of determining the struct a benzene molecule. Loschmidt hy- fan and Loschmidt, they are not properly number of atoms in a given volume, the pothesized that the six carbon atoms be recognized for the experimental data they name associated with this number (actually connected in a ring with one hydrogen produced to verify the kinetic theory of a close relative of the number) is Avo- atom be attached to each of the six carbon gases. Loschmidt’s work on the diffusion gadro, who merely postulated that a given atoms. Since there was no experimental ev- of gases and Stefan’s on the diffusion of volume of gas is proportional to the num- idence to confirm this structure, his dis- heat gave Maxwell the impetus to expand ber of atoms or molecules. Usage of the covery was largely ignored and forgotten, and expound his kinetic theory. He wrote Loschmidt number is limited to German except by one August Kekule.´ Four years first, “Professor Loschmidt, of Vienna, has speaking countries. Poor Loschmidt never after reading Loschmidt’s booklet, Kekule´ recently made a series of most valuable and seemed to receive the credit he rightly proposed the same shape. This inspiration accurate experiments on the interdiffusion deserves. supposedly came to Kekule´ after having a of gases ... these results I consider to be dream where he saw snakes grabbing on The name of Loschmidt’s colleague, the most valuable hitherto obtained as data to their own tails, then whirling around in Stefan, appears in a range of areas10. He is for the construction of a molecular theory a circle. Today, it is very common to hear most well-known for his discovery of the of gases.”13 He then followed, “Prof. Ste- about Kekule’s´ dream and his “discovery” T 4 radiation law. It was widely known that fan of Vienna, has recently, by a very del- of the structure of the benzene molecule. radiation was not proportional to the tem- icate method, succeeded in determining of http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 8 Volume 3, Issue 2, Autumn 2008 ESI NEWS

John Crepeau is at the Department of Me- 7Alfred Bader, “Loschmidt’s Graphical Formu- the conductivity of air, and he finds it, as chanical Engineering, University of Idaho, lae of 1861”, in Pioneering Ideas for the Physical he tells us, in striking agreement with the 1776 Science Center Drive, Idaho Falls, and Chemical Sciences: Josef Loschmidt’s Contribu- 14 tions and Modern Development in Structural Organic value predicted by the [kinetic] theory.” Idaho 83402, USA. Chemistry, W. Fleischhacker and T. Schonfeld,¨ eds. (New York and London: Plenum, 1997), pp. 65-80. Despite being 8James Clerk Maxwell, “On Loschmidt’s Experi- in close contact for ments on Diffusion in Relation to the Kinetic Theory most of their pro- References of Gases”, Nature, 8 (1873), pp. 298-300. 1 ductive years, these Ludwig Boltzmann, “Josef Stefan,” Populare¨ 9S. E. Virgo, “Loschmidt’s Number”, Science two friends and col- Schriften, Translated by Kevin B. Homer, (Leipzig: Progress, 27 (1933), pp. 634-649. Barth, 1905) pp. 92-103. 10 leagues have since 2 John Crepeau, “Josef Stefan: His life and legacy Ludwig Boltzmann, “In Remembrance of Josef in the thermal sciences”, Experimental Thermal and been recognized Populare¨ Schriften Loschmidt”, , Translated by Dustin Fluid Science, 31 (2007), pp. 795-803. in vastly different Marshall, (Leipzig: Barth, 1905), pp. 228-252. 11Janez Strnad, “On Stefan’s radiation law”, re- 3Janez Strnad, Jozef Stefan: The Centenary of His ways. searcher: Journal of Research and Innovation in Death, (Ljubljana, 1993). Slovenia, 28 (3) (1998). This article was 4Alfred Bader and Leonard Parker, “Joseph 12Bozidar Sarler, “Stefan’s work on solid-liquid based in part on a presentation made at Loschmidt, Physicist and Chemist”, Physics Today, 54 (2001), pp. 45-50. phase changes”, Engineering Analysis with Boundary the ESI in May 2008. An expanded version Elements 5Stephen M. Stigler, “Stigler’s Law of Eponymy”, , 16 (1995), pp. 83-92. will be published in an upcoming issue of Science and Social Structure, (New York: New York 13James Clerk Maxwell, “On Loschmidt’s Experi- Physics in Perspective. Academy of Sciences, 1980), pp.147-157. ments on Diffusion in Relation to the Kinetic Theory 6Josef Loschmidt, “Konstitutionformeln der of Gases”, Nature, 8 (1873), pp. 298-300. The author gratefully acknowledges the Organische Chemie in Graphischer Darstellung”, 14James Clerk Maxwell, “Molecules”, Nature, 8 help and hospitality of Wolfgang L. Reiter. Chemische Studien I. (1873), pp. 437-441.

Remarks on Boltzmann’s from Dresden, with Mach in the back- (“Bilder”). In that respect, Boltzmann was ground as their somewhat reluctant god- critically following Heinrich Hertz (1857– Philosophy father regarding “energetics”. The ener- 1894) and his conception of pictures pre- Wolfgang L. Reiter geticists’ positivistic view and phenomeno- sented in the preface of his Principles of logical (Machian) epistemology discard- Mechanics.5Although we have no direct ing atomism had their complement in (written) evidence by Boltzmann himself it Although Boltzmann’s Pierre Duhem’s (1861–1916) positivis- seems to be plausible to assume that he al- writings on the philos- tic approach of thermodynamics. Further- ready had been acquainted with the notion ophy of science and more, Henri Poincare´ (1854–1912) was not of mental pictures or representations much epistemology belong a keen supporter of atomism. So, during the earlier through the work of Robert von last decade of the 19th century the scien- Zimmermann (1824–1898) and his text- to his lesser-known 6 and little-discussed tific community of continental Europe with book Philosophische Propadeutik¨ . Boltz- legacy, he made impor- Germany and France leading, turned away mann studied philosophy with Zimmer- tant contributions to this field, presenting from atomism and philosophical materi- mann who became professor at the Univer- a theory of scientific change that was in- alism in favour of phenomenological and sity of Vienna in 1861. positivistic approaches. The support of the spired by Darwin’s theory of evolution. He Hertz – Darwin – Mach even speculated on an extension of physical kinetic theory and atomism was restricted to supporters in England and the Nether- theory to biology, recognising that there is Boltzmann admired the work of Hertz but lands. no contradiction between biological evolu- had a different view on the relation of tion and the laws of thermodynamics.1 For At the 67. Versammlung der Gesell- the concept of reality and reality proper. the famous 10th (1902) edition of the En- schaft der Deutschen Naturforscher und Boltzmann criticised the existence of a cyplopaedia Britannica Boltzmann wrote Arzte¨ in Lubeck¨ in September 1895 Ost- priori valid laws of logic or knowledge, an article on “Model”, extending his earlier wald, Helm and Boltzmann used the op- or explanations independent of our expe- writings on pictures.2 portunity to bring their arguments before rience. Following Charles Darwin (1809– 3 1882) Boltzmann’s evolutionist conception In Defense of Atomism the German scientific community. Boltz- mann most successfully fought for atom- of epistemology did not allow for such laws Around the time when Boltzmann moved ism and the kinetic theory during a two-day of knowledge. to Leipzig, both the concept and the con- debate with Ostwald and Helm and he in- What then will be the po- sistency of atomism were challenged on spired a younger generation of physicists, sition of the so-called laws of physical as well as philosophical grounds. among them Max Planck (1858–1947) and thought in logic? Well, in the The reversibility and recurrence paradoxes Arnold Sommerfeld (1868–1951). Boltz- light of Darwin’s theory they and the problem of specific heat weak- mann, the “bullish” defender of his stand- will be nothing else but inher- ened the credibility of atomistic theories. point (so Sommerfeld in his often cited ited habits of thought.7 Moreover, Boltzmann’s atomism was put report on the fierce debate),4 was not a into question by the “energetics“ doctrines philosophical doctrinist but rather flexible This passage in his polemics against represented by Ostwald and the theoret- when he was reflecting on the construc- Schopenhauer of 1905 sounds like the pro- ical physicist Georg Helm (1851–1923) tion of representations (models) or pictures grammatic statement of one of the fol-

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 20089 lowers of evolutionary epistemology in the In his lecture on the “Development of mechanistic (realistic) world view. On the eighties of the last century. Darwin was a methods of theoretical physics“ delivered epistemological level he is cautious enough strong ally for his mechanistic, i.e. materi- at the 71.Versammlung der Gesellschaft not to identify his mechanical atoms with alistic world view: der Deutschen Naturforscher und Arzte¨ in the real world out there. Munich in September 1899 Boltzmann re- In my view all salvation for marks: Uber¨ die Beschaffenheit philosophy may be expected der Atome aber wissen wir to come from Darwin’s the- [...] namely that no the- noch gar nichts und wer- ory. As long as people believe ory can be objective, actu- den auch solange nichts wis- in a special spirit that can ally coinciding with nature, sen, bis es uns gelingt, aus cognize objects without me- but rather that each theory is den durch die Sinne beobacht- chanical means, or in a spe- only a mental picture of phe- baren Tatsachen eine Hy- cial will that likewise is apt to nomena, related to them as pothese zu formen. 12 will that which is beneficial to sign is to designatum. From us, the simplest psychological this it follows that it cannot be And he prophetically remarks in this essay phenomena defy explanation. our task to find an absolutely of 1886: correct theory but rather a pic- Merkwurdigerweise¨ ist Only when one admits that ture that is, as simple as pos- hier am ersten wieder von der spirit and will are not some- sible and that represents phe- Kunst Erfolg zu hoffen, welche thing over and above the body nomena as accurately as pos- 9 sich auch bei Erforschung der but rather the complicated ac- sible. Himmelskorper¨ so machtig¨ tion of material parts whose And he concluded: erwies, von der Spektralanal- ability so to act becomes in- yse.13 creasingly perfected by devel- The question whether mat- opment, only when one ad- ter consists of atoms or is con- mits that intuition, will and tinuous reduces to the much New Physics self-consciousness are merely clearer one, whether [the con- the highest stages of de- ception of enormously many His farsighted consideration of spectral velopment of those physico- individuals or that of] the con- analysis as a powerful tool to reveal the chemical forces of matter by tinuum is able to furnish a bet- inner structure of atoms, “...die Beschaf- which primeval protoplasmic ter picture of phenomena.10 fenheit der Atome...“, of 1886 is in sharp bubbles were enabled to seek contrast to the observation that he never Moreover, Boltzmann’s ontological po- regions that were more and mentioned the phenomena of radioactiv- sition regarding atoms was flexible and he avoid those that were less ity discovered twelve years later. This is repeatedly made clear that he preferred an favourable for them, only then puzzling because as a member of the Vi- open and pragmatic attitude. In his discus- does everything become clear enna Academy of Sciences he was well sion on the identity of psychic processes in psychology.8 informed about the seminal role Vienna with certain material processes in the brain played in fostering radioactivity research For Boltzmann strong philosophical con- Boltzmann adds a most remarkable end- in Paris in supplying the Curies in 1898– victions, like Hertz’ apriorism or Mach’s note: 99 with more then one ton of pitch- phenomenology, were of little use in That is, if the concept blend residues from the then Austrian physics. Boltzmann in a very pragmatic of continuum is properly un- uranium mine in St. Joachimsthal (now way was led by the power of physical derstood, an interplay of its Jachimov, Czech Republic), which en- models of explanation (his “Bilder“) jus- atoms, by which of course abled Marie Curie (1867–1934) and Pierre tified by mathematical consistency. What we must not imagine mate- Curie (1859–1906) to discover polonium Boltzmann probably had in mind when he rial points but perhaps vec- and radium. Moreover, with their work in was referring to pictures or models was tors or whatever. Nor do the 1899, Stefan Meyer (1872–1949) and Egon an “isomorphism“ between the structural atoms necessarily have to be von Schweidler (1873–1948), both Boltz- elements of a physical proposition and immutable.11 mann’s students at the Vienna institute, the attributed mathematical elements but correctly distinguished between the radia- strongly guided (or modeled) by pictures Theories of the continuum had been re- tion from radium (α-rays) and from polo- or visualizable representations. Opposing garded in Boltzmann’s times as phe- nium (β-rays) by their different behaviour Mach’s epistemological anti-realism and nomenological theories in opposition to in a magnetic field. (The electromagnet had empiricism Boltzmann’s position was that atomistic theories and hence atomism been available at Boltzmann’s institute.) of a realist. But his epistemological realism was not a phenomenological theory. What They proved that the deflection of the ra- was not a naive realism equating physical Boltzmann is telling us here in defence of dium rays was identical to that of cathode models (“Bilder“) with direct representa- atoms is the fact that continuum mechan- rays, that is, that the radium rays consist of tions of reality. One even is tempted to find ics — if “properly understood“ — also negatively charged particles. This led to the traits of an instrumentalism in Boltzmann’s has to go beyond pure phenomenology by fundamental insight of the corpuscular na- writings when he presents the atomistic hy- assuming its own “atoms“, “perhaps vec- ture of these rays which is strong experi- pothesis as the most convincing and com- tors or whatever“. On the methodological mental support of the atomistic nature of prehensive, simple and elegant description level Boltzmann’s mechanical atom is rep- these new phenomena.14 Although Boltz- of the natural phenomena. resented by a picture (“Bild“) central to his mann wrote a popular account on X–rays http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 10 Volume 3, Issue 2, Autumn 2008 ESI NEWS a few month after their discovery by Kon- The gigantic structures Scheibe, Die Philosophie der Physiker. (ref. 33), rad Rontgen¨ (1854–1923),15 he never men- of the Brooklyn Bridge that pp.105-106. 5 tioned the ongoing work on radioactivity stretches beyond sight and the Heinrich Hertz, Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt. Gesammelte research at his own institute. Probably he Eiffel tower that soars with- Werke, Band III. (Leipzig: Johann Ambrosius Barth, took notice of the recent developments and out end rest not only on the 1894), pp. 1-5; reprinted in Ostwalds Klassiker Vol. just did not comment on them, but it seems solid framework of wrought 263. (Frankfurt am Main: Verlag Harri Deutsch, 2. Auflage 1996.) more likely that he had lost interest in ac- iron, but on the solider one of 6Robert Zimmermann, Philosophische 17 tual physical research during his last years. elasticity theory. Propadeutik¨ . (Wien: Wilhelm Braumuller,¨ 1852, 2nd In his Lectures on Natural Philosophy 1903 ed. 1860, 3rd ed. 1867.) See also Andrew D. Wilson, – 1906 he did not present anything refer- In 1894, at the 66. Versammlung der “Hertz, Boltzmann and Wittgenstein Reconsidered.“ Studies in History and Philosophy of Science, 20 ring to new physics. During the last decade Gesellschaft der Deutschen Naturforscher und Arzte¨ in Vienna he gave a lecture (1989) No. 2, 245-263. of his life he no longer contributed to the 7 ¨ Ludwig Boltzmann, “Eine These Schopen- forefront of research but commented on “On Airship Flight“ (“Uber Luftschif- hauers“. in Populare¨ Schriften. (ref. 10), pp. 385- what he had achieved and tried to secure fahrt“), on flying objects heavier than air 402; eingeleitet und ausgewahlt¨ von Engelbert Broda (Braunschweig und Wiesbaden: Friedr. Vieweg & his work in writing monographs and popu- by demonstrating models of aeroplanes of the Austrian aeronautics pioneer Wil- Sohn, 1979), pp. 240-257, on page 252; translated into lar articles. English in Brian McGuinness (ed.), Ludwig Boltz- helm Kress (1836–1913) which Boltzmann mann. Theoretical Physics and Philosophical Prob- Throughout his life Boltzmann was let fly around in the congregation hall of lems. Selected Writings (Dordrecht: Reidel, 1974), pp. very interested in technology. He consid- the assembly at the Vienna Musikverein.18 185-198, on page 194. 8 ered technological progress as a confirma- Boltzmann was correct in his clear-sighted ibid., in Populare¨ Schriften. (ref. 10), pp. 385- 402; eingeleitet und ausgewahlt¨ von Engelbert Broda tion of the sciences. prediction that not airships but aeroplanes (ref. 36), pp. 240-257, on page 251; translated into En- would be the superior technology of the glish in Brian McGuinness (ed.), (ref. 36), pp. 185- That is why I do not regard future. 198, on page 193. 9“[...] dass keine Theorie etwas Objektives, mit technological achievements as der Natur wirklich sich Deckendes sein kann, dass unimportant by-products of Wolfgang L. Reiter, Vice President of the vielmehr jede nur ein geistiges Bild der Erscheinung natural science but as a logi- ESI, is honorary professor for history of ist, das sich zu diesem verhalt,¨ wie das Zeichen zum cal proof. Had we not attained science at the Faculty for Historical and Bezeichneten. Daraus folgt, dass es nicht unsere Auf- gabe sein kann, eine absolut richtige Theorie, sondern these practical achievements, Cultural Studies of the University of Vi- vielmehr ein moglichst¨ einfaches, die Erscheinungen we should not know how to in- enna. moglichst¨ gut darstellendes Abbild zu finden.“ Lud- fer. Only those inferences are wig Boltzmann, “Uber¨ die Entwicklung der Methoden correct that lead to practical der theoretischen Physik in neuerer Zeit“. in Populare¨ 16 References Schriften. (ref. 10), pp. 198-277; eingeleitet und aus- success. gewahlt¨ von Engelbert Broda (ref. 36), pp. 120-149, 1Stephen G. Brush, Ludwig Boltzmann and the on page 137; translated into English in Brian McGuin- In fact Boltzmann argues here as a Foundation of Natural Science. in Ilse Maria Fasol- ness (ed.), (ref. 36), pp. 77-100, on page 90. 10 scientific realist (and atomist) implicitly Boltzmann and Gerhard Ludwig Fasol, ed., Ludwig “Die Frage, ob die Materie atomistisch zusam- Boltzmann (1844-1906). Zum hundertsten Todestag mengesetzt oder ein Kontinuum ist, reduziert sich against idealism (and phenomenological (Wien, New York: Springer-Verlag, 2006), pp. 65-80; auf die viel klarere, ob die Vorstellung enorm vieler anti-atomism) in taking the progress of sci- on p. 80. Einzelwesen oder die eines Kontinuums ein besseres ence (and technology) as a confirmation 2Ludwig Boltzmann, Model. Encyclopaedia Bri- Bild der Erscheinungen zu liefern vermoge.“¨ Lud- tannica. Vol. XXX, pp. 788-791, (London: “The wig Boltzmann, “Uber¨ die Entwicklung der Methoden of his epistemological position – realism. Times” Printing House, 1902); reprinted in: Cam- der theoretischen Physik in neuerer Zeit“. in Populare¨ So, the growing experience and the ac- bridge University Press, 11th edition, 1911; Brian Schriften. (ref. 10), pp. 198-277; eingeleitet und aus- cumulation of increasingly refined knowl- McGuinness (ed.), (ref. 36), pp. 211-220. gewahlt¨ von Engelbert Broda (ref. 36), pp. 120-149, edge produced by science (and applied by 3Walter Hoflechner,¨ Ludwig Boltzmann: Leben on page 138; translated into English in Brian McGuin- und Briefe (Graz: Akademische Druck- u. Ver- ness (ed.), (ref. 36), pp. 77-100, on page 91.Insertion technology) corroborate realism. Despite lagsanstalt, 1994), pp. I 164-171. Erhard Scheibe, in brackets [...] is added by the author. of Boltzmann’s strong anti-metaphysical Die Philosophie der Physiker (Munchen:¨ C. H. Beck, 11 “D. h. bei richtiger Auffassung des Begriffs des sentiments his own concept of realism is 2006), pp. 104-119. Christa Jungnickel and Russell Kontinuums ein Spiel der Atome desselben, worunter part of a metaphysical concept. Probably, McCormack, Intellectual Mastery of Nature. Theoret- man sich freilich nicht materielle Punkte denken muss, ical Physics from Ohm to Einstein. Vol. 2. The Now sondern vielleicht Vektoren oder wer weiss was. Auch Boltzmann was quite aware of that fact Mighty Theoretical Physics 1870-1925. (Chicago and mussen¨ die Atome nicht notwendig unveranderlich¨ and therefore disguised his epistemologi- London: The University of Chicago Press, 1986), pp. sein.“ Ludwig Boltzmann, “Uber¨ die Frage nach der cal realism as a methodological principle, 217-227. objektiven Existenz der Vorgange¨ in der unbelebten 4 in sharp contrast to Mach’s concept. What “Das Referat fur¨ die Energetik hatte Helm - Natur“. in Populare¨ Schriften. (ref. 10), pp. 162-187; Dresden; hinter ihm stand Wilhelm Ostwald, hin- eingeleitet und ausgewahlt¨ von Engelbert Broda (ref. I mean by disguising realism is Boltz- ter beiden die Naturphilosophie des nicht anwe- 36), pp. 94-119, on page 112; translated into English mann’s view of the conception of pictures senden Ernst Mach. Der Opponent war Boltzmann, in Brian McGuinness (ed.), (ref. 36), pp. 57-76, on (“Bilder“) as representations of the “world sekundiert von Felix Klein. Der Kampf zwischen page 76. 12 out there“. Admittedly, since Boltzmann Boltzmann und Ostwald glich, ausserlich¨ und in- Ludwig Boltzmann, “Der zweite Hauptsatz der nerlich, dem Kampf des Stieres mit dem geschmei- mechanischen Warmetheorie“.¨ in Populare¨ Schriften. never has developed his philosophy of sci- digen Fechter. Aber der Stier besiegte diesmal den (ref. 10), pp. 25-50, on p. 30; eingeleitet und aus- ence in a systematic manner, his ideas are a Torero trotz all seiner Fechtkunst. Die Argumente gewahlt¨ von Engelbert Broda (ref. 36), pp. 26-46, on profound source of various interpretations Boltzmann’s schlugen durch. Wir damals jungeren¨ page 31; translated into English in Brian McGuin- Mathematiker standen auf der Seite Boltzmann’s; es ness (ed.), (ref. 36), pp. 5713-32, on page 17; here the and misunderstandings. war uns ohne weiters einleuchtend, dass aus der translation is highly misleading. What makes reading his Populare¨ einen Energiegleichung unmoglich¨ die Bewegungsgle- 13Ibid., the passage cited here in German is missing Schriften such a delightful experience is ichungen auch nur eines Massenpunktes, geschweige in the English translation. 14 ¨ Boltzmann’s humour. In a slightly poetic denn eines Systems von beliebigen Freiheitsgraden St. Meyer and E. v. Schweidler, “Uber das gefolgert werden konnten.”¨ Arnold Sommerfeld, Das Verhalten von Radium und Polonium im magnetis- manner Boltzmann speaks about the power Werk Boltzmann’s. Lecture at the occasion of Lud- chen Felde,“ Anzeiger der kaiserl. Akad. d. Wiss., of theory. wig Boltzmann’s 100th birthday. Quoted in Erhard mathem.- naturw. Kl. 22 (November 3, 1899), 1-4.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 2008 11

dus 130 (1900), 809; cf. also E. Rutherford, Radioac- 17Der Riesenbau der Brooklyner Brucke,¨ welche St. Meyer and E. v. Schweidler, “Uber¨ das Verhalten tive Transformations (New Haven: Yale University sich unabsehbar in die Lange,¨ und der des Eifel- von Radium und Polonium im magnetischen Felde. Press, 1906), p. 9. For the history of early experiments turms, der sich endlos in die Hohe¨ erstreckt, sie (I. Mitteilung),“ Physikalische Zeitschrift 1 (1899), of the deflection of Becquerel radiation, see Marjorie beruhen nicht bloss auf dem festen Gefuge¨ des 90-91. St. Meyer and E. v. Schweidler, “Weitere No- Malley, The Discovery of the Beta Particle, American Schmiedeeisens, sondern auf dem festeren der Elas- tizen uber¨ das Verhalten von Radium im magnetis- Journal of Physics 39 (1971), 1454-1461. tizitatstheorie.¨ Ludwig Boltzmann, “Uber¨ die Bedeu- chen Felde,“ Anzeiger der kaiserl. Akad. d. Wiss., 15Ludwig Boltzmann, “Rontgens¨ neue Strahlen“ in tung von Theorien“. in Populare¨ Schriften. (ref. 10), mathem.- naturw. Kl. 23 (November 9, 1899), 1- 2. Populare¨ Schriften. (ref. 10), pp. 188-197. pp. 76-80; eingeleitet und ausgewahlt¨ von Engelbert St. Meyer and E. v. Schweidler, “Uber¨ das Verhalten 16“Deshalb halte ich die Errungenschaften der Broda (ref. 36), pp. 54-58, on page 56; translated into von Radium und Polonium im magnetischen Felde. Technik nicht fur¨ nebensachliche¨ Abfalle¨ der Natur- English in Brian McGuinness (ed.), (ref. 36), pp. 33- (II. Mitteilung),“ Physikalische Zeitschrift 1 (1899), wissenschaften, ich halte sie fur¨ logische Beweise. 36, on page 35. 113-114. F.O. Giesel, “Uber¨ die Ablenkbarkeit der Hatten¨ wir diese praktischen Errungenschaften nicht 18Ludwig Boltzmann, ”Uber¨ Luftschiffahrt“, in Becquerelstrahlen im magnetischen Felde,“ Annalen erzielt, so wussten¨ wir nicht, wie man schliessen Populare¨ Schriften, (ref. 10), pp. 81-91. As a mere cu- der Physik 69 (1899), 834-836. J. Elster und H. Geitel, muss. Nur solche Schlusse,¨ welche praktischen Erfolg riosity I note, that Ludwig Wittgenstein (1889–1951) “Uber¨ den Einfluss eines magnetischen Feldes auf die haben, sind richtig“. Ludwig Boltzmann, “Eine These who considered Boltzmann’s philosophical writings durch die Becquerelstrahlen bewirkte Leitfahigkeit¨ Schopenhauers“. in Populare¨ Schriften. (ref. 10), pp. as an important influence on his own thinking first der Luft,“ Verhandlungen der Deutschen Physikalis- 385-402; eingeleitet und ausgewahlt¨ von Engelbert studied mechanical engineering at the Technische chen Gesellschaft 1 (1899), 136-138 (Sitzung von 5. Broda (ref. 36), pp. 240-257, on page 249; translated Hochschule in Charlottenburg, Berlin for two years Mai 1899). H. Becquerel, “Deviation´ du rayonnement into English in Brian McGuinness (ed.), (ref. 36), pp. before he went to Manchester where he studied aero- du radium dans un champ electrique,“´ Comptes Ren- 185-198, on page 193. nautics.

http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 12 Volume 3, Issue 2, Autumn 2008 ESI NEWS

ESI News

Walter Thirring: Autobiography New ESI Lectures in Mathematics and Physics

Walter Thirring, founding father and Hon- Boltzmann’s Legacy orary President of the ESI, has written an autobiography. The book is on the market G. Gallavotti, W. L. Reiter, J. Yngvason (eds.) in December this year. Zurich:¨ European Mathematical Society Publishing House 2008.

Walter Thirring, Lust am Forschen This book contains lectures presented – Lebensweg und Begegnungen. Wien: at the International Symposium “Boltz- Seifert Verlag 2008. manns Legacy“ held at the ESI in June 2006 to commemorate the 100th an- niversary of Ludwig Boltzmann’s death in Duino. New Members of the ESI Society 2008: The text covers a broad spectrum of Markus Arndt (Faculty of Physics, University of Vienna) topics ranging from equilibrium and Reinhard Burger¨ (Faculty of Mathematics, University of Vienna) nonequilibrium statistical physics, er- Joachim Hermisson (Faculty of Mathematics, University of Vi- godic theory and chaos to basic ques- enna) tions of biology and historical accounts of Boltzmann’s work. Besides the lectures presented at the sympo- sium the volume also contains contributions specially written for New activity in history and philosophy of science: this occasion. The articles give a broad overview of Boltzmann’s Jaques Bouveresse (College de France) legacy to the sciences from the standpoint of some of present day’s gave a lecture on ”Ludwig Boltzmann und leading scholars in the field. das Problem der Erklarung¨ in der Wis- The book addresses students and researchers in mathematics, senschaft” on Oktober 29, 2008 at the Boltz- physics and the history of science. mann Lecture Hall of the ESI. This lecture was preceded by a lecture on ”Musil als Philosoph” at the Kleiner Festsaal at the main building of the University of Vienna. News from the Scientific Community The lecture was organized in cooperation with the Institute for Markus Arndt Philosophy, University of Vienna, and the Institute Vienna Circle by E. Nemeth, W. L. Reiter and F. Stadler. (Speaker of the group Quantum Optics, Quantum Physics and Quantum Informa- tion of the Faculty of Physics, Univer- International Workshop ”Routes to Mauthausen”, November sity of Vienna), member of the ESI soci- 27 to November 29, 2008. ety, was awarded the Wittgenstein Price 2008. The Workshop was orga- nized by Gerhard Botz Arndt already has won the START-Price of the (Professor for Contempo- FWF 2001. rary History, Faculty for Historical and Cultural Markus Aspelmeyer (Institut for Quantum Optics and Quantum Studies, University of Vi- Information - IQOQI, OAW¨ Vienna), Massimo Fornasier (Jo- enna and Ludwig Boltz- hann Radon Institute for Computational and Applied Mathemat- mann Institute for Histori- ics, OAW¨ Linz) and Daniel Grumiller (Institut fur¨ Theoretische cal Social Science). The workshop was opened by a Round Table Physik, Vienna University of Technology) had been awarded the Discussion on ”Humanity and the Experience of Violence and Start Price of the FWF 2008. Genocide” on November 27, 2008 at the premisses of the ESI.

All friends of the ESI are cordially invited to a Christmas Party at the Institute on Thursday, December 18, 2008, 5.00 p.m.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 2008 13 Current and Future Activities of the ESI

Thematic Programmes 2008 Entanglement and correlations in many-body quantum mechanics, August 18 – October 17, 2009 Combinatorics and Statistical Physics, February 1 – June 15, 2008 Organizers: B. Nachtergaele, F. Verstraete and R. Werner Organisers: M. Bousquet-Melou, M. Drmota, C. Krattenthaler, B. Nienhuis The dbar-Neumann problem: analysis, geometry and potential theory, October 27 - December 24, 2009 Workshop, May 25 – June 7, 2008 Organizers: F. Haslinger, B. Lamel, E. Straube Summer School, July 7 – July 18, 2008

Metastability and Rare Events in Complex Systems, February 1 Thematic Programmes 2010 – April 30, 2008 Organizers: P.G. Bolhuis, C. Dellago, E. van den Eijnden Quantitative Studies of Nonlinear Wave Phenomena, January Workshop, February 17 – February 23, 2008 7 - February 28, 2010 Organizers: P.C. Aichelburg, P. Bizon, W. Schlag Hyberbolic Dynamical Systems, May 12 – July 5, 2008 Organisers: H. Posch, D. Szasz, L.-S. Young Quantum field theory on curved space-times and curved target-spaces, March 1 - April 30, 2010 Workshop, June 15 – June 29, 2008 Organizers: M. Gaberdiel, S. Hollands, V. Schomerus, J. Yngvason Operator Algebras and Conformal Field Theory, August 25 – December 15, 2008 Matter and radiation, May 3 - July 30, 2010 Organisers: Y. Kawahigashi, R. Longo, K.-H. Rehren, J. Yngvason Organizers: V. Bach, J. Frhlich, J. Yngvason Thematic Programmes 2009 Topological String Theory, Modularity and Non-Perturbative Physics, June 7 - August 15, 2010 Representation theory of reductive groups — local and global Organizers: L. Katzarkov, A. Klemm, M. Kreuzer, D. Zagier aspects, January 2 – February 28, 2009 Organizers: G. Henniart, G. Muic and J. Schwermer Anti - de Sitter holography and the quark-gluon plasma: analytical and numerical aspects, August 2 - October 29, 2010 Mathematics at the Turn of the 20th Century: Explorations and Beyond, January 7 - 12, 2009 Organizers: A. Rebhan, K. Landsteiner, S. Husa Organizers: D.D Fenster, J. Schwermer Higher Structures in Mathematics and Physics, August 15 - November 15, 2010 Number theory and physics, March 1 - April 18, 2009 Organizers: A. Alekseev, H. Bursztyn, T. Strobl Organizers: A. Carey, H. Grosse, D. Kreimer, S. Paycha, S. Rosenberg and N. Yui

Gravity in Three Dimensions, April 14 - 24, 2009 Organizers: H. Grosse, D. Grumiller, R. Jackiw, D. Vassilevich

Selected topics in spectral theory, May 4 – July 25, 2009 Organizers: B. Helffer, T. Hoffmann-Ostenhof and A. Laptev

Catalysis from First Principles, May 25 - 30, 2009 Organizers: J. Hafner, J. Norskov, M. Scheffler

Large cardinals and descriptive set theory, 2 weeks in June – July 2009 Organizers: S. Friedman, M. Goldstern, R. Jensen, A. Kechris and W.H. Woodin http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 14 Volume 3, Issue 2, Autumn 2008 ESI NEWS Topics in Mathematical Physics, July 21 – July 31, 2008 Other Scientific Activities in 2008 Organizers: C. Hainzl, R. Seiringer and J. Yngvason

Tensor network methods and entanglement in quantum Vienna Central European Seminar on Particle Physics and many-body systems, January 16 – January 18, 2008 Quantum Field Theory, November 28 – November 30, 2008. Organizers: F. Verstraete, G. Vidal and M. Wolf The topic of the Seminar is ”Highlights in Computational Quantum Field Theory” and is opened by a public lecture on ”Quarks, Gluons, and Lattices” by Michael Creutz, BNL, on Ab-initio density-functional studies of intermetallic November 28, 2008. compounds, January 23 – January 25, 2008 This Semininar, organized by the Faculty of Physics, University Organizer: J. Hafner of Vienna, is supported by the ESI. Organizer: H. Huffel¨ 15th Anniversary of the ESI, April 14, 2008 Organizers: W.L. Reiter, K. Schmidt, J. Schwermer and J. Supersymmetry and Noncommutative Quantum Field Yngvason Theory. In memoriam Julius Wess, December 4 - 6, 2008. Organizers: H. Grosse, P. Schupp Frontiers in Mathematical Biology: Mathematical population genetics, April 14 – April 18, 2008 Profinite Groups, December 7 – December 20, 2008 Organizers: R. Burger¨ and J. Hermisson Organizers: K. Auinger, F. Grunewald, W. Herfort and P.A. Zalesski

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 3, Issue 2, Autumn 2008 15

Erwin Schrodinger¨ Lectures Autumn Term 2008/09

The Erwin Schrodinger¨ Lectures are directed towards a general audience of mathematicians and physicists. In particular it is an intention of these lectures to inform non-specialists and graduate students about recent developments and results in some area of mathematics or mathematical physics. These lectures take place in the Boltzmann Lecture Room of the ESI. Each lecture will be followed by an informal reception at the Common Room of the ESI.

Vaughan Jones (UC Berkeley): Flatland, a great pleasure to do Jean-Pierre Serre (College de France, Paris): Variation with p of algebra. Nov 4, 2008 the number of solutions (mod p) of polynomial equations. Dec 11, 2008 Matthias Kreck (Hausdorff Research Institute, Bonn): Codes and James W. Cogdell (Ohio State University, Columbus): On sums 3-dimensional manifolds. Nov 13, 2008 of three squares. Jan 22, 2009

Organizer: J. Schwermer

Senior Research Fellows Lecture Courses Autumn and Spring Term 2008/09

To stimulate the interaction with the local scientific community, the ESI offers lecture courses on an advanced graduate level. These courses are taught by Senior Fellows of the ESI, whose stays in Vienna are financed by the University of Vienna, the Vienna University of Technology, and the Austrian Federal Ministery for Education, Science and Culture. These courses take place in the Erwin-Schrodinger¨ Lecture Room of the ESI.

Nigel Higson (Penn State University): Index Theory, Groupoids Goran Muic (University of Zagreb): Selected Topics in the The- and Noncommutative Geometry Monday and Wednesday, 14:00- ory of Automorphic Forms for Reductive Groups Thursday, 13:00- 16:00, Friday 10:30-12:30, starting on November 24, 2008 15:00, starting on October 16, 2008 Michel Loss (Georgia Tech): Inequalities April 15 - June 30, 2009

Feng Xu (University of California, Riverside): Operator Algebras Raimar Wulkenhaar (Munster):¨ Spektraltripel in der nichtkom- and Conformal Field Theory Wednesday and Friday, 16:00-18:00, mutativen Geometrie und Quantenfeldtheorie March 1 - June 15, starting on November 4, 2008 2009

Mathematics at the Turn of the 20th Century: Explorations and Beyond January 7 - 12, 2009

This workshop aims to bring together scholars from a variety of fields with a common interest in the mathematical sciences of the 19th and 20th centuries in their historical context. Special attention will be given to include young participants. The programme will combine lectures on recent results with ample time for informal discussions and collaborations. In commemoration of Hermann Minkowski’s death on January 12, 1909, the talks given on Monday, January 12, 2009 will illuminate Minkowski’s work in mathematics and physics.

Among these talks are: Samuel J. Patterson (Gottingen):¨ The number-theorist Hermann Scott Walter (Nancy): Hermann Minkowski and theoretical Minkowski physics in Gottingen¨

Organizers: D. D Fenster, J. Schwermer

http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 16 Volume 3, Issue 2, Autumn 2008 ESI NEWS

Editors: Wolfgang L. Reiter, Klaus Schmidt, Joachim Schwer- ESI Contact List: mer, Jakob Yngvason Administration Contributors: Isabella Miedl: [email protected] John Crepeau: [email protected] Scientific Directors Christoph Dellago: [email protected] Joachim Schwermer: [email protected] Gerhard Ecker: [email protected] Jakob Yngvason: [email protected] Walter Grimus: [email protected] Wolfgang L. Reiter: [email protected] President Klaus Schmidt: [email protected]

The ESI is supported by the Austrian Federal Ministry for Science and Research ( ). The ESI Senior Research Fellows Programme is additionally supported by the University of Vienna.

This newsletter is available on the web at: ftp://ftp.esi.ac.at/pub/ESI-News/ESI-News3.2.pdf

IMPRESSUM: Herausgeber, Eigentumer¨ und Verleger:INTERNATIONALES ERWIN SCHRODINGER¨ INSTITUTFUR¨ MATHEMATISCHE PHYSIK, Boltzmanngasse 9/2, A-1090 Wien. Redaktion und Sekretariat: Telefon: +43-1-4277-28282, Fax: +43-1-4277-28299, Email: [email protected] Zweck der Publikation: Information der Mitglieder des Vereins Erwin Schrodinger¨ Institut und der Offentlichkeit¨ in wissenschaftlichen und organisatorischen Belangen. Forderung¨ der Kenntnisse uber¨ die mathematischen Wissenschaften und deren kultureller und gesellschaftlicher Relevanz.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/