Time Reversibility and Ergodicity Snook Prize Topics for 2013 and 2014 Wm. G. Hoover & Carol G. Hoover Ruby Valley NV USA With messages from Mike Klein & Doug Henderson Ian Snook Conference on Chemical Physics Royal Melbourne Ins8tute of Technology 4-‐5 December 2014 2013 Ian Snook Prize : Time-‐Reversible Random Numbers arXiv: 1305.0961 2014 Ian Snook Prize : Ergodicity of a Time-‐Reversibly Thermostated Harmonic Oscillator Computa8onal Methods in Science and Technology* 20(3) 87-‐92 (2014) * .pdf ﬁle available @ www.williamhoover.info What is meant by Time Reversibility ? J. A. Roberts and G. R. W. Quispel in Physics Reports (1992) : “Dynamical systems with 8me t and phase-‐space variables r are Reversible if they are invariant under the combina8on ( +t à -‐t, r à G•r ) with G an Involu8on : (G•G= Iden8ty) .” Carl Deimann showed, in arXiv 1402.7010* : with the ﬂow dq/dt = p ; dp/dt = -‐ q -‐ p and the involu8on G(q,p) = (q,-‐p-‐q)/[ q2 + (4/3)( p + (q/2) )2 ] à Reversibility ! Evidently a damped harmonic oscillator is ( RQ ) reversible ! We prefer and use a diﬀerent deﬁni&on of a reversible set of mo8on equa8ons : A reversed movie of the mo8on obeys the same equa8ons. * Communica8ons in Theore8cal Physics ( Beijing ) , 62, 521-‐540 ( 2014 ) Can we reverse “Stochas8c Brownian Dynamics” By reversing our random number generator ? func8on rund(intx,inty) i = 1029*intx + 1731 j = i + 1029*inty + 507*intx -‐ 1731 intx = mod(i,2048) j = j + (i -‐ intx)/2048 inty = mod(j,2048) rund = (intx + 2048*inty)/4194304.0 return end This two-‐seed random number generator provides 222 = 4 194 304 = 20482 numbers between 0 and 1 without gaps or repe88on. Winner of the 2013 Ian Snook Prize Professor Federico Ricci-‐Tersenghi [ University of Rome ] arXiv: 1305.1805 The inverse is even shorter than the original random number generator ! An interes8ng applica8on is the Ehrenfests’ Dog-‐Flea Model . We can reverse “Stochas8c Brownian Dynamics” By reversing our random number generator ! func,on back(intx,inty) oldx = mod(205*intx + 1497,2048) inty = inty + 4194304 -‐ 1536*oldx -‐ (1029*oldx + 1731 -‐ intx)/2048 inty = mod(205*inty,2048) intx = oldx rund = (intx + 2048*inty)/4194304.0 return end 1000 steps forward & backward What is Ergodicity* ? [ Wikipedia says : This ar,cle may be too technical for most readers to understand. ] `` We say that T is ergodic . . . ‘’ What is Quasi-‐Ergodicity+ ? ( evidently this concept is only for Hamiltonian Systems ) `` . . . The opinion is generally prevalent that from the point of view of empiricism such ques8ons are meaningless . . . ’’ *. George David Birkhoﬀ, ``Proof of the Ergodic Theorem’’ PNAS 17, 656 ( 1931 ) . +. John von Neumann, ``Proof of the Quasi-‐Ergodic Hypothesis” PNAS 18, 70 (1932) . 2014 Ian Snook Prize : Ergodicity of a Time-‐Reversibly Thermostated Harmonic Oscillator Computa8onal Methods in Science and Technology 20, 87 (2014) Nosé-‐Hoover Time-‐Reversible [ not ergodic ] Oscillator (1985) : dq/dt = p ; dp/dt = -‐q -‐ ζp ; dζ/dt = p2 -‐ 1 Martyna-‐Klein-‐Tuckerman Oscillator (1992) : dq/dt = p ; dp/dt = -‐q -‐ ζp ; dζ/dt = p2 -‐ 1 -‐ ξζ ; dξ/dt = ζ2 -‐ 1 Hoover-‐Holian Oscillator (1996) : dq/dt = p ; dp/dt = -‐q – ζp -‐ ξp3 ; dζ/dt = p2 -‐ 1 ; dξ/dt = p4 -‐ 3p2 The last two of these have been thought `` quasi-‐ergodic ’’, with The dynamics agreeing closely with the con8nuity equa8on f( r ) : f(q,p,z,x) α exp[-‐q2/2] exp[-‐p2/2]exp[-‐ζ2/2]exp[-‐ξ2/2] Our own ( informal ) deﬁni8on of the Ergodic -‐ Quasiergodic concept : If the phase-‐space ﬂow equa8on dr/dt = v , with df(r,t)/dt = -‐ ∇ . ( fv ) has a sta&onary solu8on [ such as f( r ) = f( q,p ) = e[ -‐H( q,p )/kT ] ] , the ﬂow equa8on is Ergodic -‐ Quasiergodic if the long-‐8me trajectory comes arbitrarily close to each phase-‐space point where f( r ) is nonzero . Five Phase-‐Space Flows, Four Ergodic and One Not ß No Field But recent work by Puneet Patra and Baidurya Bhaiacharya * à that the Martyna-‐Klein-‐Tuckerman oscillator is not ergodic. Because the equilibrium solu8on is known, a four-‐dimensional Gaussian, there are a variety of tests which can be made, many of which were carried out by Martyna, Klein, and Tuckerman and ourselves as well as Patra and Bhaiacharya, but with opposite conclusions as to MKT ergodicity . The 2014 Ian Snook Prize ( $500.00 US ) will be awarded to the author(s) of the most convincing paper addressing this ques8on before January 2015 . *. `` Non-‐Ergodicity of Nosé-‐Hoover Chain Thermostat in Computa&onally Achievable Time ”, published in Physical Review E in October 2014 . There are also some interes8ng fractal structures when there is a temperature gradient, T = 1 + εtanh(q)* : p(q) ε = 0.42 p(q) ε = 0.38 Conserva8ve and
Structures can coexist in the qp phase space . * with Clint Sproi , arXiv : 1401.1762: [ Physical Review E (2014) ] There are also some interes8ng fractal structures when there is a temperature gradient, T = 1 + εtanh(q)* : p(q) p(q) HH ε = 0.40 MKT ε = 0.20 These are double Poincaré sec8ons with both ζ and ξ equal to 0 . * with Clint Sproi, arXiv: 1401.1762: [ Physical Review E (2014) ] Martyna – Klein – Tuckerman Chain Thermostat!
( q, p, -1, +1 ) section!
Double Poincaré Sec8on 4
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“ The [ MKT ] thermostat therefore does not generate the canonical distribu,on or preserve quasi-‐ergodicity for the Poincaré Sec,on ”. Puneet Kumar Patra and Baidurya Bhaiacharya, “ Nonergodicity of the Nosé-‐Hoover Chain Thermostat [ MKT Thermostat ] in Computa8onally Achievable Time ”, arXiv 1407.2353 = Physical Review E 90, 043303 ( 2014 ) 2014 Ian Snook Prize : Ergodicity of a Time-‐Reversibly Thermostated Harmonic Oscillator Computa8onal Methods in Science and Technology 20 , 87-‐92 (2014) ß 1996 2007 with Ian & Marie à Ian Snook Remembered ( with wine ) [ Mike Klein, Temple University, Philadelphia ] In the early 1980’s I spent a few weeks with Bob Wa]s at ANU but also visited with a few other like-‐minded colleagues elsewhere in Australia. One such visit involved an invited talk in Melbourne. As I recall, three characters: [ Snook, Smith, and van Megan ] met me at the airport and we immediately headed oﬀ for lunch at a BYOB Chinese restaurant. Apparently, my hosts, who came suitably armed with many excellent samples, already knew my keen interest in wine. The number of bo]les on hand was impressive and clearly intended to ensure a thorough educa,on. The predictable end result was that the ,me for my seminar passed while we were all s,ll in the restaurant! However, nobody seemed concerned and it was decided to skip the seminar in favor of con,nuing scien,ﬁc discussions at Snook’s home, where a selec,on of Australian red wines was conveniently wai,ng for me to con,nue my educa,on. As I recall, there were very many truly wonderful examples on hand and the tas,ng con,nued well into the evening at which point it was decided to start sampling Australian champagnes. Alas, I cannot remember much more about the visit to Melbourne other than waking up on a couch in the living room the following morning ! Importantly, I was not hung over and indeed was forever grateful for my thorough educa,on in Australian wines. Fond Memories of Ian Snook ( without wine ) [ Douglas Henderson, Brigham Young University, Provo Utah ] My family and I spent a memorable year (1966-‐67) in Melbourne where I collaborated with John Barker at CSIRO. Ajerwards, John and I both joined the IBM Research Laboratory in San Jose, California. While there, Farid Abraham, John Barker, and I developed the infamous Henderson-‐Abraham-‐Barker equa,on [ HAB ] for an inhomogeneous ﬂuid. We used this advance to convince the IBM management to form a sta,s,cal mechanics project. One of the ﬁrst successes of our group was to interest Ian to take a leave from RMIT and come to San Jose for a year and to convince IBM Australia to help fund the visit of Ian and his family during 1976-‐77. During this year Ian and I made one of the ﬁrst simula,ons of a dense ﬂuid near a surface and compared the results with those of the HAB equa,on. This work was the subject of a talk that I gave subsequently at a Gordon Research Conference. Ian's work and reputa,on con,nued to blossom. Some,me around 1980 John Barker and I had a friendly 'argument' about which one of us had the higher opinion of Ian. In 1999 Derek Chan arranged for me to visit the University of Melbourne for a few weeks. During this visit, Ian, Ed Smith, and I had a very pleasant lunch together at a restaurant near RMIT. Ed, and now Ian, have moved on to a be]er place. For me, the world seems emp,er as a result. The Last email we got from Ian was dated 9 March 2013 : “ I saw this at the town of Dover which is very far South in Tassie just a short 8me ago when we were staying there . A very beau8ful part of the world when it is not raining , which it wasn’t , but a very strange commandment is shown on the aiached : ` NO ASCENDING Any8me Day or Night ! ’ ‘’ ‘