Quasi-Static Hydrodynamic Limits and their Large Deviations Stefano - PowerPoint PPT Presentation

Quasi-Static Hydrodynamic Limits and their Large Deviations Stefano Olla CEREMADE, Universit´ e Paris-Dauphine - PSL Supported by ANR LSD INPAM Bedlewo, April 14, 2018

The problem with thermodynamics Every mathematician knows that it is impossible to understand any elementary course in thermodynamics. V.I. Arnold, Contact Geometry: the Geometrical Method of Gibbs’s Thermodynamics. (1989)

Thermodynamics concerns Macroscopic Objets at Macroscopic Time Scale Vapor machine of Joseph Cugnot (1770)

A simple thermodynamic system A one dimensional system (rubber under tension): Mechanical Equilibrium: L = L( τ ) , τ = tension

A simple thermodynamic system A one dimensional system (rubber under tension): Mechanical Equilibrium: L = L( τ ) , τ = tension Thermodynamic Equilibrium L = L( τ,θ ) θ is the temperature

A simple thermodynamic system A one dimensional system (rubber under tension): Mechanical Equilibrium: L = L( τ ) , τ = tension Thermodynamic Equilibrium L = L( τ,θ ) θ is the temperature Empirical definition of temperature.

▸ There exists a family of thermodynamic equilibrium states, parametrized by certain extensive or intensive variables. – lenght (volume) L and energy U (extensive) ▸ In a thermodynamics transformation, ∆ U = W + Q W : mechanical work done by the force ¯ τ , Q : energy exchanged with the heat bath ( Heat ).

Quasi-Static Transformations Existence of thermodynamic processes where the system is always at some equilibrium. These processes are described by continuous curves on the space of parameters. This way we can define isothermal lines and adiabatic lines etc.

Quasi-Static Transformations Existence of thermodynamic processes where the system is always at some equilibrium. These processes are described by continuous curves on the space of parameters. This way we can define isothermal lines and adiabatic lines etc. We can consider this as a hidden principle of thermodynamics.

Quasi-Static Transformations Existence of thermodynamic processes where the system is always at some equilibrium. These processes are described by continuous curves on the space of parameters. This way we can define isothermal lines and adiabatic lines etc. We can consider this as a hidden principle of thermodynamics. But, quoting Zemanski, Every infinitesimal in thermodynamics must satisfy the requirement that it represents a change in a quantity which is small with respect to the quantity itself and large in comparison with the effect produced by the behavior of few molecules.

Thermodynamic transformations and Cycles ▸ reversible or quasi-static tranformations: W = ∮ τ dL = − Q

Special quasi-static transformations ▸ Isothermal: System in contact with a thermostat while the external force τ is doing work: / W = τ d L = τ ( ∂ L ∂τ ) d τ = − d / Q + dU d θ

Special quasi–static Transformations / Q = 0. ▸ Adiabatic: d / W = τ d L = dU d

Special quasi–static Transformations / Q = 0. ▸ Adiabatic: d / W = τ d L = dU d d τ d L = − ∂ L U ∂ τ U

Carnot Cycles Q B A C D Q A → B , C → D isothermal B → C , D → A adiabatic W = ∮ τ d L = Q h − Q c = −∮ d / Q

2nd Principle and Entropy Q B A C D Q − Q c 0 = Q h T h T c / Q / Q d T = 0 , dS = d ∮ T .

Quasi-Static Isothermal Hydrodynamic Limit Slowly changing tension: N ( p 2 H N ( t ) = 2 + V ( r i )) + ¯ τ ( t ) q N ∑ i i = 1 plus random collisions with particles of the heat bath : at independent random times p i ( t ) � → ˜ p j ∼ N( 0 , T )

Isothermal Dynamics ⎧ dr i ( t ) = n 2 + α ( p i ( t ) − p i − 1 ( t )) dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dp i ( t ) = n 2 + α ( V ′ ( r i + 1 ( t )) − V ′ ( r i ( t ))) dt ⎪ ⎪ − n 2 + α γ p i ( t ) dt + n 1 + α / 2 √ ⎪ ⎨ 2 γβ − 1 dw i ( t ) , i = 1 ,.., N − 1 ⎪ ⎪ ⎪ ⎪ dp n ( t ) = n 2 + α ( ¯ τ ( t ) − V ′ ( r n ( t ))) dt ⎪ ⎪ − n 2 + α γ p n ( t ) dt + n 1 + α / 2 √ ⎪ ⎪ ⎪ ⎪ 2 γβ − 1 dw n ( t ) . ⎩ { w i ( t )} i : n -independent Wiener processes, τ ( t ) . α > 0 for quasistatic driving from ¯

Isothermal Dynamics The generator of the process is given by ∶ = n 2 + α (A ¯ L ¯ + γ S n ) , τ ( t ) τ ( t ) (1) n n where A ¯ τ n is the Liouville generator n ( p i − p i − 1 ) ∂ r i + n − 1 ( V ′ ( r i + 1 )− V ′ ( r i )) ∂ p i +( ¯ τ − V ′ ( r n )) ∂ p n (2) ∑ ∑ n = A ¯ τ i = 1 i = 1 while S n is the operator S n = n ( β − 1 ∂ 2 p i − p i ∂ p i ) ∑ (3) i = 1

Canonical Gibbs measures √ 2 πβ − 1 ∫ e − β ( V ( r ) − τ r ) dr ] . G( τ,β ) = log [ The stationary measure is given by n n d µ τ,β = e − β (E j − τ r j ) − G( τ,β ) dr j dp j = g ( n ) ∏ ∏ dr j dp j τ,β j = 1 j = 1

Canonical Gibbs measures √ 2 πβ − 1 ∫ e − β ( V ( r ) − τ r ) dr ] . G( τ,β ) = log [ The stationary measure is given by n n d µ τ,β = e − β (E j − τ r j ) − G( τ,β ) dr j dp j = g ( n ) ∏ ∏ dr j dp j τ,β j = 1 j = 1 r ( τ,β ) = β − 1 ∂ τ G( τ,β ) = ∫ r j d µ τ,β U ( τ,β ) = − ∂ β G( τ,β ) = ∫ V ( r j ) d µ τ,β + 1 2 β = ∫ E j d µ τ,β S ( U , r ) = inf τ,β > 0 {− βτ r + β U + G( τ,β )} Entropy F ( r ,β ) = sup τ { τ r − β − 1 G( τ,β )} Free Energy τ ( r ,β ) = ∂ r F ( r ,β ) .

Irreversible Isothermal Transformations α = 0: diffusive space-time scale, G ( i / N ) r i ( t ) � → 0 G ( y ) r ( y , t ) dy N ∑ 1 1 N →∞ ∫ i ∂ t r ( t , y ) = ∂ yy τ ( r ( t , y ) ,β ) , ∂ y r ( t , y )∣ y = 0 = 0 , τ ( r ( 1 , t ) ,β ) = ¯ τ ( t ) . In the diffusive time scale, there is a need of infinite time to converge to the new thermodynamic equilibrium, and this is an irreversible non-quasistatic trasformation.

Quasi-Static Isothermal Hydrodynamic Limit (Letizia-Olla, AOP 2018, De Masi-Olla JSP 2015) For α > 0, for all t > 0 n G ( i / n ) r i ( t ) � → r ( t ) ∫ 0 G ( x ) dx 1 ∑ 1 n →∞ ¯ n i = 1 r ( t ) = r ( β, ¯ τ ( t )) = β − 1 ( ∂ τ G)( β, ¯ τ ( t )) is the where we denote ¯ τ ( t ) . equilibrium volume at temperature β − 1 and tension ¯

Quasi-Static Isothermal Hydrodynamic Limit (Letizia-Olla, AOP 2018, De Masi-Olla JSP 2015) For α > 0, for all t > 0 n G ( i / n ) r i ( t ) � → r ( t ) ∫ 0 G ( x ) dx 1 ∑ 1 n →∞ ¯ n i = 1 r ( t ) = r ( β, ¯ τ ( t )) = β − 1 ( ∂ τ G)( β, ¯ τ ( t )) is the where we denote ¯ τ ( t ) . equilibrium volume at temperature β − 1 and tension ¯ Similar result with macroscopic profile of temperatures β ( y ) n G ( i / n ) r i ( t ) � → 0 G ( x ) ¯ r ( t , y ) dx 1 1 ∑ n →∞ ∫ n i = 1 r ( t , y ) = r ( β ( y ) , ¯ τ ( t )) : quasistatic non-equilibrium with ¯ transformations .

Proof of isothermal QS limit t ( r 1 , p 1 ,..., r n , p n ) the density of the distribution at time t with f n respect to µ ¯ τ ( t ) ,β : ∂ t ( f n τ ( t ) ,β ) = (L ¯ t ) g n τ ( t )∗ t g n f n n ¯ τ ( t ) ,β ¯ H n ( t ) = ∫ f n H n ( 0 ) = 0 t log f n t d µ ¯ τ ( t ) ,β ,

Proof of isothermal QS limit t ( r 1 , p 1 ,..., r n , p n ) the density of the distribution at time t with f n respect to µ ¯ τ ( t ) ,β : ∂ t ( f n τ ( t ) ,β ) = (L ¯ t ) g n τ ( t )∗ t g n f n n ¯ τ ( t ) ,β ¯ H n ( t ) = ∫ f n H n ( 0 ) = 0 t log f n t d µ ¯ τ ( t ) ,β , ( ∂ p i f n t ) 2 dt H n ( t ) = − n 2 + α γβ − 1 ∫ n d ∑ d µ c , n f n ¯ τ ( t ) ,β i = 1 t − β ¯ τ ′ ( t ) ∫ n ( r i − ¯ r ( t )) f n ∑ t d µ c , n τ ( t ) ,β . ¯ i = 1

proof of isothermal QS limit By entropy inequality, for any λ > 0 small enough n τ ′ ( t ) ∫ ( r i − ¯ r ( t )) f n ∑ t d µ c , n β ¯ ¯ τ ( t ) ,β i = 1 τ ( t ) ,β + λ − 1 H n ( t ) ≤ λ − 1 log ∫ e λβ ¯ τ ′ ( t ) ∑ n r ( t )) d µ c , n i = 1 ( r i − ¯ ¯ ≤ λ Cn + λ − 1 H n ( t ) ,

proof of isothermal QS limit By entropy inequality, for any λ > 0 small enough n τ ′ ( t ) ∫ ( r i − ¯ r ( t )) f n ∑ t d µ c , n β ¯ τ ( t ) ,β ¯ i = 1 τ ( t ) ,β + λ − 1 H n ( t ) ≤ λ − 1 log ∫ e λβ ¯ τ ′ ( t ) ∑ n r ( t )) d µ c , n i = 1 ( r i − ¯ ¯ ≤ λ Cn + λ − 1 H n ( t ) , dt H n ( t ) ≤ λ − 1 H n ( t ) + λ Cn , d and since H n ( 0 ) = 0, it follows that H n ( t ) ≤ e t / λ λ Ct n . This is not yet what we want to prove but it implies that ( ∂ p i f n t ) 2 n T τ ( t ) ,β dt ≤ ∑ C ∫ ∫ d µ c , n n 1 + α . f n ¯ 0 i = 1 t

proof of isothermal QS limit ( ∂ p i f n t ) 2 n τ ( t ) ,β dt ≤ T ∑ C ∫ ∫ d µ c , n n 1 + α . f n ¯ 0 i = 1 t This gives only information on the distribution of the velocities. Uning entropic hypocoercive bounds we have the same for ( ∂ q i f n t ) 2 n τ ( t ) ,β dt ≤ T C ∑ ∫ ∫ d µ c , n n 1 + α . f n ¯ 0 i = 1 t where ∂ q i = ∂ r i − ∂ r i + 1 . and this is enough to prove that n ( r i − ¯ r ( t )) f n τ ( t ) ,β � → 0 , 1 ∑ ∫ t d µ c , n ¯ n i = 1 i.e. H n ( t ) � → 0 . n