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Density fluctuations and transport in normal and supercooled quantum liquids

Density fluctuations and transport in normal and supercooled quantum liquids. Eran Rabani School of Chemistry Tel Aviv University. Outline. Quantum Mode-Coupling Theory Quantum generalized Langevin equation (QGLE) Quantum mode-coupling approximations Density fluctuations Classical liquids

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Density fluctuations and transport in normal and supercooled quantum liquids

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  1. Density fluctuations and transport in normal and supercooled quantum liquids Eran Rabani School of Chemistry Tel Aviv University

  2. Outline • Quantum Mode-Coupling Theory • Quantum generalized Langevin equation (QGLE) • Quantum mode-coupling approximations • Density fluctuations • Classical liquids • Quantum liquids • Quantum supercooled liquids • Quantum Kob-Andersen model • Preliminary results • Experimental realization

  3. Literature • E. Rabani and D.R. Reichman, Phys. Rev. E 65, 036111 (2002) • E. Rabani and D.R. Reichman, J. Chem. Phys.116, 6271 (2002) • D.R. Reichman and E. Rabani, J. Chem. Phys.116, 6279 (2002) • E. Rabani and D.R. Reichman, Europhys. Lett.60, 656 (2002) • E. Rabani and D.R. Reichman, J. Chem. Phys.120, 1458 (2004) • E. Rabani and D.R. Reichman, Ann. Rev. Phys. Chem.56, 157-185 (2005) • E. Rabani, K. Miyazaki, and D.R. Reichman, J. Chem. Phys.122, 034502 (2005) Collaboration: Kunimasa Miyazaki, ColumbiaUniversity David Reichman, ColumbiaUniversity Roi Baer, Hebrew University Daniel Neuhauser, UCLA Financial Support: Israel Science Foundation, EU-STERP, Ministry of Science, US-Israel Binational Science Foundation

  4. Theory

  5. Quantum mode coupling • Step 1:Formulation of an exact quantum generalized Langevin equation (QGLE) using Zwanzig-Mori projection operator technique, for theKubo transformof the dynamical variable of interest: • Step 2:Approximate memory kernel for the QGLE using aquantum mode-couplingtheory. • Step 3:Solution of the QGLE with the approximate memory kernel combined with exact static input generated from a suitable PIMCscheme.

  6. Density fluctuations To study density fluctuations we need to specify the dynamical variable and the corresponding correlation function (the intermediate scattering function): The exact QGLE for the Kubo transform of the intermediate scattering function is given by The formal expression for the memory kernel is

  7. Binary portion The formal expression for the memory kernel is Short time expansion to second order in time The time moments are given in terms of the density moments:

  8. where the new projection operator projects onto the following slow modes: Mode coupling portion The projected dynamics is replaced with the full dynamics projected onto the slow decaying modes: In addition, four point correlation functions are replaced by a product of two point correlation functions:

  9. Total memory kernel

  10. PIMC scheme We need to calculate the following static Kubo transforms: where Using the coordinate representation of the matrix element: We obtain (to lowest order in e using P Trotter slices): Our result looks similar to the Barker energy estimator, however, it is numerically less noisy.

  11. Classical liquids

  12. Liquid lithium The normalized intermediate scattering function for liquid lithium. Theredcurves are results obtained from molecular dynamics simulations and thebluecurves are results obtained from a classical mode-coupling theory. The agreement between the theory and simulations is remarkable for all q values shown.

  13. Quantum liquids

  14. Quantum liquids p-H2 The normalized intermediate scattering function for liquid para-hydrogen.Theredcurves are results obtained from the QMCT and thegreencurves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT.

  15. Dynamic structure factor p-H2 The normalized dynamic structure factor for liquid para-hydrogen.Red– QMCT.Green- MaxEnt.Black– QVMassuming a single relaxation time.Bluecircles- experimental results F. J. Bermejo, B. Fak, S. M. Bennington, R. Fernandez-Perea, C. Cabrillo,J. Dawidowski, M. T. Fernandez-Diaz, and P. Verkerk, Phys. Rev. B 60, 15154 (1999).

  16. Quantum liquids o-D2 The normalized intermediate scattering function for liquid ortho-deuterium.Theredcurves are results obtained from the QMCT and thegreencurves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT.

  17. Dynamic structure factor o-D2 The normalized dynamic structure factor for liquid ortho-deuterium.Red– QMCT.Green - MaxEnt.Black– QVM assuming a single relaxation time.Bluecircles- experimental results from M. Mukherjee, F. J. Bermejo, B. Fak, and S. M. Bennington, Europhys. Lett.40, 153 (1997).

  18. Quantum Transport

  19. QGLE for VACF We need to obtain a QGLE for thevelocity autocorrelation function(v is the velocity of a tagged liquid particle along an arbitrary direction): • Following similar lines to those sketched for the classical theory, we obtain an exactquantum generalized Langevin equation (QGLE): where we have used the following projection operator and the memory kernel is formally given by

  20. Quantum Mode Coupling Theory The Kernel is approximated by Fast decaying quantum binary term: The slow decaying quantum mode-coupling term: The vertex:

  21. MC Memory Kernel for VACF The slow decaying quantum mode-coupling term is obtained using a set of approximations. The projected dynamics is replaced with the full dynamics projected onto the slow decaying modes: where the new projection operator is given by: In addition, four point correlation functions are replaced by a product of two point correlation functions:

  22. Static input from PIMC The static input for the memory kernel of the velocity autocorrelation function generated from a PIMC simulation method for liquidpara-hydrogenat T=14K (redcurve) and T=25K (bluecurve).

  23. Velocity autocorrelation function The normalized velocity autocorrelation function calculated from the quantum mode-coupling theory (bluecurve) and from an analytic continuation of imaginary-time PIMC data (bluecurve) for liquidpara-hydrogenat T=14K (lower panel) and T=25K (upper panel). The good agreement between the two methods is a strong support for the accuracy of the quantum mode-coupling approach for liquidpara-hydrogen.

  24. Memory kernel for VACF The Kubo transform of the memory kernel for the velocity autocorrelation function for liquid para-hydrogen at T=14K (upper panel) and T=25K (lower panel). Shown are the fast-decaying binary term (redcurve), the slow-decaying mode-coupling term (greencurve) and the total memory kernel (bluecurve). The contribution of the slow mode-coupling portion of the memory kernel is significant at the low temperature, while at the high temperature, the kernel can be approximated by only the fast binary portion.

  25. Self-Diffusion - Liquid para-H2 The frequency dependent diffusion constant for liquidpara-hydrogenat T=14Kand T=25K. The self-diffusion obtained from the Green-Kubo relation is0.30and1.69(Å2/ps) for T=14K and T=25K, respectively. These results are in good agreement with the experimental results (0.40 and 1.60) and with the maximum entropy analytic continuation method (0.28and1.47).

  26. VACF – Liquid ortho-D2 The normalized velocity autocorrelation function and its Kubo transform calculated from the quantum mode-coupling theoryfor liquid ortho-deuterium (upper panel) and liquid para-hydrogen (lower panel) at T=20.7K

  27. Self-Diffusion - Liquid ortho-D2 The frequency dependent diffusion constant for liquidpara-hydrogen (greencurve) and ortho-deuterium (redcurve) at T=20.7K.The self-diffusion obtained from the Green-Kubo relation is0.49and0.64(Å2/ps) for para-hydrogenand ortho-deuterium, respectively. The result for ortho-deuterium is in reasonable agreement with the experimental results (0.36Å2/ps).

  28. Normal Liquid Helium The normalized velocity autocorrelation function calculated from the quantum mode-coupling theory (bluecurve),from an analytic continuation of imaginary-time PIMC data (redcurve), and from a semiclassical approach (Makri – green curve) for liquid helium above the l transition.

  29. Self-Diffusion – Normal Liquid Helium The frequency dependent diffusion constant for normal liquid heliumat T=4K. The results shown were calculated from the quantum mode-coupling theory (bluecurve),from an analytic continuation of imaginary-time PIMC data (redcurve), from a semiclassical approach (Makri – green curve), and from the CMD method (black curve)

  30. Quantum glasses

  31. Quantum glasses • Can we form a structural quantum glass (onset of quantum fluctuations, super fluidity)? • Are there anythermodynamicsignatures that are different for a quantum glass? • Are there anydynamicsignatures that are different for a quantum glass?

  32. Kob-Andersen model The Kob-Andersen model is based on a binary mixture of Lennard-Jones (BMLJ) particles with the following parameters: The system undergoes anergodic-to-nonergodictransition atT=0.435. Classical mode-coupling theory predicts a transition at aboutT=0.92. Kob and Andersen, Phys. Rev. E 51, 4626 (1995) Kob and Andersen, Phys. Rev. E 52, 4134 (1995) Nauroth and Kob, Phys. Rev. E 55, 657 (1997)

  33. Classical results for the Kob-Andersen model

  34. Early b Late b aregime Intermediate Scattering Function MCT predictions Self intermediate scattering function versus time for A and B particles at two wave length for several temperatures. Kob and Andersen, Phys. Rev. E 52, 4134 (1995).

  35. Self-diffusion Left: Mean square displacement versus time forAparticles at different temperatures. Right: Diffusion constant versus temperature forAandBparticles.Solidlines are best fits to power-law anddashed lines are best fits to Vogel-Fulcher law.

  36. Quantum results for theKob-Andersen model

  37. Average potential energy Average potential energy per particle for the quantum Kob-Andersen model. Simulation were done forN=500and P=100. There is a clear change near T=1 (similar to the classical case). Is there any slowing down near this temperature?

  38. Intermediate scattering function Intermediate scattering function atr=1.2for several values of the temperature atq=qmax. No significant slowing down is observed far below the classicalTc=0.92.

  39. Static structure factor

  40. Pair correlation function

  41. Low wave vector results Intermediate scattering function atr=1.2for several values of the temperature atq=qmax/2. TheAparticles show coherent fluctuation not observed classically at this value of q.

  42. And even lower Intermediate scattering function atr=1.2for several values of the temperature atq=qmax/4.

  43. Experimental realization

  44. T=10 Mixtures of p-H2 and o-D2

  45. T=8 Mixtures of p-H2 and o-D2

  46. T=8, TP density T=10, TP density T=6, < TP density Centroid configurations At the triple-point (TP) density the mixture freezes into an ordered crystals below T=10. But at a slightly lower temperature, the system can be supercooled and even at T=6 is still disordered.

  47. Classical Results – Kob-Andersen Time dependence of the coherent and incoherent intermediate scattering function for two wave vectors at T = 2. The dashed line with the symbols are the results from the simulation and the solid lines are the prediction of the classical MCT theory. From Kob and collaborators (Phys. Rev. E 55, 657 (1997), J. Non-Cryst. Solids 307, 181 (2002)).

  48. Classical Results – Kob-Andersen Time dependence of the coherent and incoherent intermediate scattering function for two wave vectors at T = 0.466. The dashed line with the symbols are the results from the simulation and the solid lines are the prediction of the classical MCT theory. From Kob and collaborators (Phys. Rev. E 55, 657 (1997), J. Non-Cryst. Solids 307, 181 (2002)).

  49. Simple Example - VACF Now, lets make asimple Gaussianapproximation to the memory kernel: Even this simple approximation (short time expansion) captures some of the hallmarks of normal monoatomic liquids. The reason is that the approximation is done at the level of the memory kernel, and thus better results are obtained for the correlation function itself. However, this approximation completely neglects the long time decay of the memory kernel.

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