Universal Quantum Transport in Chaotic Cavities and Painlevé Transcendents
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1 XXIII Marian Smoluchowski Symosium on Statistical Physics, Kraków, Se 27, 2010 Alied Mathematics 38 Phys. Rev. Lett. 101, (2008) & JPA 42, (2009) Universal Quantum Transort in Chaotic Cavities and Painlevé Transcendents Eugene Kanzieer 1,2 & Vladimir Al Osiov 3 1 Deartment of Alied H.I.T. - Holon Institute of Technology 2 Deartment of Physics of Comlex Weizmann Istitute of Science 3 Fakultät für Universität Duisburg-Essen Thanks Satya Majumdar The Israel Science Foundation
2 Alied Mathematics 37 Outline The system, the roblem, and the main result S. Oberholzer Quantum chaotic cavity Statistics of the Landauer conductance Paul Painlevé Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics
3 Alied Mathematics 36 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions Gaston Darboux Satya Majumdar
4 Alied Mathematics 35 The system, the roblem The system, the roblem cavity leads couling Left reservoir Right reservoir S. Oberholzer, Universität Bazel Landauer conductance
5 Alied Mathematics 34 The system, the roblem The system, the roblem cavity leads couling Universal transort regime Electron dwell time much larger than the Ehrenfest time Left reservoir Right reservoir RMT dynamics: Landauer conductance
6 Alied Mathematics 33 The system, the roblem, and the main result The system, the roblem, and the main result th cumulant of the Landauer conductance Fifth Painlevé transcendent!! BPC Universal transort regime Electron dwell time much larger than the Ehrenfest time Left reservoir Right reservoir RMT dynamics: Landauer conductance
7 Alied Mathematics 32 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions
8 Alied Mathematics 31 Painlevé roerty: Brief excursion Brief excursion into Painlevé roerty Rational function of its arguments E. Picard (1889) All movable singularities are restricted to oles (no movable branch oints) (a) linear 2nd order DEs (b) Weierstrass DE (c) Riccati DE [ Painlevé equations P 6 I P VI ] nonlinear secial functions P. Painlevé (1900,1902) B. Gambier (1905) R. Fuchs (1910)
9 Alied Mathematics 30 Painlevé roerty: Brief excursion Brief excursion into Painlevé roerty σp III σp V Hamiltonian system Bäcklund transformations [ Painlevé equations P 6 I P VI ] nonlinear secial functions P. Painlevé (1900,1902) fascinating B. Gambier (1905) R. roerties Fuchs (1910) P. Clarkson, Painleve equations nonlinear secial functions, J. Com. Al. Math. 153, 127 (2003)
10 Alied Mathematics 29 Painlevé roerty: Brief excursion Brief excursion into Painlevé roerty Remark: Painlevé roerty is the child of «ure reason» Question: Should Painlevé transcendents be relevant to the descrition of the hysical world? Answer: Yes (seven decades after Painlevé discovery)!!
11 Alied Mathematics 28 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions
12 Alied Mathematics 27 Painlevé functions in statistical hysics Painlevé transcendents in statistical hysics 2D Ising model (a) R. Peierls (1936), (b) H. Kramers & G. Wannier (1941), T. Wu, (c) B. L. McCoy, Onsager C. (1944) Tracy, and and C. E. N. Barouch Yang (1955) (1976) (a) Existence of the 2 nd order hase transition, (b) Critical temerature, (c) Sontaneous magnetisation σp III
13 Alied Mathematics 26 Painlevé functions in statistical hysics Painlevé transcendents in statistical hysics Imenetrable Bose gas T. Esslinger ETH Zürich M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) M. Girardeau (1960) A. Lenard (1964) σp V
14 Alied Mathematics 25 Painlevé functions in statistical hysics Painlevé transcendents in statistical hysics Growth models in (1+1)D / oriented digital boiling J. Gravner, C. Tracy, and H. Widom (2001) Universal regime of shae fluctuations σp II
15 Alied Mathematics 24 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions
16 Alied Mathematics 23 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions
17 Alied Mathematics 22 Cumulants of the Landauer conductance Cumulants of the Landauer conductance (derivation) P. Brouwer cavity leads (1995) couling Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
18 Alied Mathematics 21 Cumulants of the Landauer conductance Cumulants of the Landauer conductance (derivation) BPC Scattering matrix aroach Left reservoir Right reservoir Landauer conductance
19 Alied Mathematics 20 Cumulants of the Landauer conductance Cumulants of the Landauer conductance (derivation) Cumulant generating function Truncate! Hua (1963) Zyczkowski & Sommers (2000) Itzykson-Zuber formula, but: high degeneracy of C-matrices BPC Left reservoir Right reservoir Landauer conductance
20 Alied Mathematics 19 Cumulants of the Landauer conductance Cumulants of the Landauer conductance (derivation) Cumulant generating function Truncate! Hua (1963) Zyczkowski & Sommers (2000) Itzykson-Zuber formula, but: high degeneracy of C-matrices
21 Alied Mathematics 18 Cumulants of the Landauer conductance Cumulants of the Landauer conductance (derivation) Cumulant generating function Ga formation robability in LUE Fruitful ideas of integrability How can it be evaluated nonerturbatively?!
22 Cumulants of the Landauer conductance 17 Kadomtsev-Petviashvili equation (the whole hierarchy) Virasoro constraints Relation Virasoro algebra
23 Cumulants of the Landauer conductance 16 Solve the two together at!! Kadomtsev-Petviashvili equation (the whole hierarchy) Virasoro constraints
24 Cumulants of the Landauer conductance 15 Solve the two together at!! Final result cumulant generating function Introduce Observe the fifth Painleve transcendent
25 Cumulants of the Landauer conductance 14 Solve the two together at!! Final result cumulant generating function!! Fifth Painleve transcendent Taylor exansion th cumulant of the Landauer conductance
26 Cumulants of the Landauer conductance 13 Cumulants of the Landauer conductance: Recurrence relation!! Fifth Painleve transcendent Taylor exansion th cumulant of the Landauer conductance
27 Cumulants of the Landauer conductance 12 Cumulants of the Landauer conductance: exansion!! Fifth Painleve transcendent Taylor exansion th cumulant of the Landauer conductance
28 Alied Mathematics 11 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions
29 Alied Mathematics 10 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Gaston Darboux Conclusions
30 Alied Mathematics 09 Distribution function Distribution of the Landauer conductance (derivation) Lalace transform of conductance robability density function Hankel determinant Gaston Darboux The Toda Lattice equation Æ recursive rocedure exact solution
31 Alied Mathematics 08 Distribution function Distribution of the Landauer conductance (derivation) Recurrence brings Inverse Lalace transform yields the conductance. d. f. Gaston Darboux The Toda Lattice equation Æ recursive rocedure exact solution
32 Alied Mathematics 07 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions Gaston Darboux Satya Majumdar
33 Alied Mathematics 06 Gram-Charlier exansion 1/n cumulant exansion Distribution function The end of the large-n controversy uncontrolled combination Osiov-Kanzieer result [Phys. Rev. Lett. 101, (2008)] Gaussian Exonential Power -law Bohigas-Majumdar-Vivo result [Phys. Rev. Lett. 101, (2008)] Gaussian Power -law
34 Alied Mathematics 05 Distribution function The end of the large-n controversy Painlevé analysis In the global regime fixed: Gaussian Power -law No exonential tails!!
35 Alied Mathematics 04 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Paul Painlevé Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions Gaston Darboux Satya Majumdar
36 Alied Mathematics 03 Other works Relation to other works Savin, Sommers 2006 Savin, Sommers, Wieczorek 2007, 2008 Novaes 2008 Osiov, Kanzieer 2008, 2009 Vivo, Majumdar, Bohigas 2008, 2010 Khoruzhenko, Savin, Sommers 2009
37 Alied Mathematics 02 Outline The system, the roblem, and the main result Why the result is interesting Brief excursion into the Painlevé roerty Aearance of Painlevé transcendents in statistical hysics Derivation (Landauer conductance) Cumulants of the Landauer conductance Distribution function and the end of large-n controversy Relation to other works Paul Painlevé Furher results (if time ermits) Statistics of thermal to shot noise crossover Joint cumulants of Landauer conductance and noise ower Conclusions Gaston Darboux Satya Majumdar
38 Alied Mathematics 01 Conclusions Conclusions
39 XXIII Marian Smoluchowski Symosium on Statistical Physics, Kraków, Se 27, 2010 Alied Mathematics 00 Phys. Rev. Lett. 101, (2008) & JPA 42, (2009) Universal Quantum Transort in Chaotic Cavities and Painlevé Transcendents Eugene Kanzieer 1,2 & Vladimir Al Osiov 3 1 Deartment of Alied H.I.T. - Holon Institute of Technology 2 Deartment of Physics of Comlex Weizmann Istitute of Science 3 Fakultät für Universität Duisburg-Essen
Phys. Rev. Lett. 101, (2008) Quantum Transport in Chaotic Cavities and Painlevé Transcendents
30 Phys. Rev. Lett. 101, 176804 (2008) Quantum Transort in Chaotic Cavities and Painlevé Transcendents Eugene Kanzieer 1 and Vladimir Al. Osiov 2 1 H.I.T. Holon Institute of Technology, Israel 2 Universität
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