Relativistic stable processes in quasi-ballistic heat conduction - PowerPoint PPT Presentation

Relativistic stable processes in quasi-ballistic heat conduction Samy Tindel Purdue University Purdue – 2020 Applied Mathematics Pizza Seminar Joint work with P. Chakraborty, A. Shakouri, B. Vermeersch Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 1 / 24

Outline Introduction 1 Relativistic stable processes 2 Model and data fit 3 Conclusion and perspectives 4 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 2 / 24

Outline Introduction 1 Relativistic stable processes 2 Model and data fit 3 Conclusion and perspectives 4 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 3 / 24

Experimental setting TDTR: Time domain thermoreflectance Short heat impulses Strong pump pulses get into material Weaker probe signals → in order to measure change in reflectance due to heat ֒ Transform to − V in / V out Material used: In 0 . 53 Ga 0 . 47 As Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 4 / 24

Illustration:Experimental Setting pump fs laser f mod EOM probe pump probe lock-in detection alu transducer delay line semiconductor 10X film/substrate under study detector sample Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 5 / 24

Classical heat equation Notation: We set T 0 ≡ Initial temperature distribution and ∂ t T ( t , x ) = ∂ ∂ t T ( t , x ) Equation: For x ∈ R d and d = 1 , 2 , 3 ∂ t T ( t , x ) = 1 2∆ T ( t , x ) , with T (0 , x ) = T 0 ( x ) . Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 6 / 24

Brownian motion Definition 1. Let (Ω , F , P ) probability space { W t ; t ≥ 0 } stochastic process, R -valued We say that W is a Brownian motion if: W 0 = 0 almost surely 1 Let n ≥ 1 and 0 = t 0 < t 1 < · · · < t n . The increments 2 δ W t 0 t 1 , δ W t 1 t 2 , . . . , δ W t n − 1 t n are independent For t > 0 we have 3 = e − 1 2 t ξ 2 � e ıξ W t � W t ∼ N (0 , t ) , or E Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 7 / 24

Illustration: chaotic path 1.5 1.0 0.5 Brownian motion 0.0 -0.5 -1.0 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 8 / 24

Illustration: random path 1.5 1.0 0.5 Brownian motion 0.0 -0.5 -1.0 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 9 / 24

Illustration: 2-d Brownian motion 1.5 1.0 0.5 Brownian motion 2 0.0 -0.5 -1.0 -1.5 0.0 0.5 1.0 1.5 2.0 Brownian motion 1 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 10 / 24

Feynman-Kac representation Equation: Classical heat equation ∂ t T ( t , x ) = 1 2∆ T ( t , x ) , with T (0 , x ) = T 0 ( x ) . Representation: For a Brownian motion W , T ( t , x ) = E [ T 0 ( x + W t )] Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 11 / 24

A non Brownian world A modified heat equation: Under our experimental setting The data does not match the heat equation 1 Idea: replace the Brownian motion by a Levy process 2 Levy processes: Most natural generalizations of Brownian motion 1 Involve jumps 2 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 12 / 24

Outline Introduction 1 Relativistic stable processes 2 Model and data fit 3 Conclusion and perspectives 4 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 13 / 24

Relativistic stable process Definition 2. Let m ≥ 0, 0 < α < 2 and { X m t ; t ≥ 0 } stochastic process, R -valued We say that X m is a relativistic stable process if: X m 0 = 0 almost surely 1 Let n ≥ 1 and 0 = t 0 < t 1 < · · · < t n . The increments 2 δ X m t 0 t 1 , δ X m t 1 t 2 , . . . , δ X m are independent t n − 1 t n For t > 0 we have 3 | ξ | 2 + m 2 /α � α/ 2 − m � �� �� � e ıξ X m � = exp − t E t Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 14 / 24

Remarks on relativistic stable process Case m = ∞ : We get a Brownian motion � e ıξ X m � � − c m t | ξ | 2 � E ≃ exp t Case m = 0: We get an α -stable process � e ıξ X m � = exp ( − t | ξ | α ) E t Jumps: All Levy processes have jumps Brownian motion is the only continuous Levy process Jumps can be described in terms of characteristic function α -stable have heavy tailed jumps Relativistic stable processes have light tailed jumps Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 15 / 24

Typical path of an α -stable process Examples of paths: Different values of α , for m = 0 Role of parameter α : If α is small Larger jumps Less integrability: E [ | X t | p ] < ∞ for p < α Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 16 / 24

Transition from α -stable to Brownian Theorem 3. Let X m ≡ Relativistic stable process p m t ( x ) ≡ Density of X m t Then c 1 , m t − d /α × decaying function( x ) ,  t small  p m t ( x ) = c 1 , m t − d / 2 × decaying function( x ) , t large  Interpretation: For t small, α -stable behavior For t large, Brownian behavior Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 17 / 24

Outline Introduction 1 Relativistic stable processes 2 Model and data fit 3 Conclusion and perspectives 4 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 18 / 24

Feynman-Kac representation Model: We assume, for a relativistic stable X m , T ( t , x ) = E [ T 0 ( x + X m t )] Corresponding PDE: Nonlocal, of the form ∂ t T ( t , x ) = L m T ( t , x ) , with − ∆ + m 2 /α � α/ 2 . L m = m − � Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 19 / 24

Experimental setting (repeated) TDTR: Time domain thermoreflectance Short heat impulses Strong pump pulses get into material Weaker probe signals ֒ → in order to measure change in reflectance due to heat Transform to − V in / V out Estimation for α : Experimental: α = 1 . 695 Theoretical Physics: α = 1 . 75 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 20 / 24

Data fit Comparison data/theoretical: For different values of f mod 4 f mod = 18MHz 3.5 lock-in ratio signal -V in /V out 15.1MHz 3 11.5MHz 2.5 5.36MHz 9.6MHz 2 1.8MHz 0.82MHz 1.5 1 0.05 0.1 0.3 1 3 pump-probe delay [ns] Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 21 / 24

Outline Introduction 1 Relativistic stable processes 2 Model and data fit 3 Conclusion and perspectives 4 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 22 / 24

Conclusions Advantages of the Levy formalism Natural extension of the Brownian formalism Justified by theoretical Physics considerations Connexions to a rich mathematical literature ◮ Identification of the distribution ◮ Kernel estimates Excellent data fit Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 23 / 24

Perspectives Multilayer setting: Two layers of different materials, either in 2-d or 3-d Coupling of 2 nonlocal PDEs Boundary conditions: TBD heat source Material 1 Material 2 Samy T. (Purdue) Relativistic stable processes Pizza Seminar 2020 24 / 24