Once Upon A Time, There Was A Certain Ludwig

Transcription

1 Once Upon A Time, There Was A Certain Ludwig Statistical Mechanics: Ensembles, Distributions, Entropy and Thermostatting Srinivas Mushnoori Chemical & Biochemical Engineering Rutgers, The State University of New Jersey 2 nd March 2016

2 First, a few things It is impossible to cover this topic comprehensively in fifteen minutes A lot of the math has been omitted for accessibility Have an open mind

3 Statistical Mechanics Thermodynamics has a certain reputation Classical thermodynamics especially so Statistical Thermodynamics is slightly easier to visualize (Far easier to imagine a bunch of balls flying around in a box) It is concerned with Emergent Behaviour

4 What are they? Ensembles

5 What are they? Ensembles

6 What are they? Ensembles

7 What are they? Ensembles

8 What are they? Ensembles

9 What are they? Ensembles

10 Ensembles What are they? Four distinct (and permissible) microstates

11 Ensembles What are they? Four distinct (and permissible) microstates Repeat for all atoms and all possible displacements So an ensemble is really an ensemble of states

12 Ensembles In terms of Thermodynamic Constraints: Canonical (N, V, T): Closed system Microcanonical (N, V, E): Isolated system Grand Canonical (μ, V, T): Open system Therefore, constraints define an ensemble but do not constitute it

13 The Notion of Time Another look at the four microstates: No global order of precedence can be established

14 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

15 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

16 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

17 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

18 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

19 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf Several throws later

20 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

21 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf?!

22 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

23 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf

24 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! E M

25 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! E e x M

26 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! E P i = Ce Ei/kT M

27 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! P i = Ce Ei/kT E M C = 1/ i=1 e Ei/kT M

28 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! E P i = e Ei/kT σ M i=1 e Ei/kT M

29 Distribution of Energy States Consider a thought experiment: You and a really big bookshelf The Boltzmann Distribution! E P i = e Ei/kT σ M i=1 e Ei/kT M

30 Partition Function Q (N, V, T): M Q = i=1 e Ei/kT The Partition Function A measure of the total number of accessible states Also tells us how the energy, in a sense, is partitioned between the various microstates (density of states), hence the name

31 Practical Significance of the partition Function The Partition Function can be used to compute thermodynamic properties, for example the Internal Energy E: E defined as the probabilistically weighted ensemble average energy < E >= σ M i=1 p i E i = Q(N, V, T) i=1 Some simple differentiation yields: M Ei e Ei/kT E = kt 2 lnq T

32 Boltzmann and the Second Law of Thermodynamics Boltzmann formulated the Second Law in the famous equation: S = k lnw S=Entropy k=boltzmann Constant W=Number of microstates corresponding to a given macrostate

33 Boltzmann and the Second Law of Thermodynamics Revisiting the Partition Function: M Q = i=1 e Ei/kT A low density of states means that there are literally fewer states for the system to jump to, meaning a lower entropy In the previously discussed example, the Q works out to zero, meaning that it is in the state of least entropy.

34 Thermostatting in MD simulations Thermostatting: The method of maintaining a specified temperature in the simulation Non-trivial. Must preserve thermodynamics, i.e. the correct distribution of microstates in the ensemble. Must preserve realistic dynamics, i.e. Newton s equations of motion should be able to accurately predict transport properties.

35 Thermostatting in MD simulations The meaning of temperature as seen earlier: Temperature is a direct consequence of velocities of the particles in a system Most intuitive way of controlling temperature?

36 Thermostatting in MD simulations The meaning of temperature as seen earlier: Temperature is a direct consequence of velocities of the particles in a system Most intuitive way of controlling temperature? Change velocities to see a corresponding change in temperature

37 Thermostatting: Velocity Rescaling Velocity Rescaling: controlling (scaling) the velocity distribution by a specific factor so as to arrive at the desired temperature. Velocities scaled according to the equation V new = T 0 ΤT V old Here, T 0 is the desired temperature, and T is the instantaneous temperature calculated from the velocities of the particles.

38 Thermostatting: Velocity Rescaling Velocity Rescaling however fails due to one major flaw: Consider how frequently the scaling is factored in. In the limit of scaling occurring at every timestep, energy remains constant, i.e. no fluctuations

39 Thermostatting: Velocity Rescaling Velocity Rescaling however fails due to one major flaw: Consider how frequently the scaling is factored in. In the limit of scaling occurring at every timestep, energy remains constant, i.e. no fluctuations Why is this bad?

40 The Langevin Thermostat The Langevin Thermostat is a superior thermostat to the Velocity Rescaling Method. Assumes that the particles being simulated exist in a continuum of smaller particles. Langevin Dynamics are extremely useful to mimic solvent effects in implicit solvent systems. Langevin Dynamics can rigorously reproduce the canonical ensemble.

41 The Langevin Thermostat The Langevin Thermostat accounts for the presence of a solvent by assuming a continuum of smaller particles. The smaller particles provide kicks to the particles being simulated the same way solvent molecules would. The technique simulates the effect of the solvent on the particles being simulated. Solvent hydrodynamics, however, are not captured.

42 The Langevin Thermostat The Langevin equation is very difficult to derive in three dimensions, but reasonably simple in a simple 1 dimensional system. We can start with the Newton s Equation of Motion: m dv dt = ζv = 6πηv Easy enough to solve: v t = e ζtτ m v(0) HUGE problem: < v 2 > k B T/m

43 The Langevin Thermostat The Random Kick Correction: m dv dt = ζv + δr(t) Random Force : δr(t) This is actually called a Gaussian White Noise Two properties must be fulfilled for a variable to be called a Gaussian White Noise signal <δr(t)> = 0 (Averages to zero) <δr(t) δr(t )> = 2Bδ(t t ) (Autocorrelation function)

44 The Langevin Thermostat The Random Kick Correction: m dv dt = ζv + δr(t) Random Force : δr(t) This is actually called a Gaussian White Noise Two properties must be fulfilled for a variable to be called a Gaussian White Noise signal <δr(t)> = 0 (Averages to zero) <δr(t) δr(t )> = 2Bδ(t t ) (Autocorrelation function)

45 Fluctuation-Dissipation Theorem Random kicks(i.e. fluctuations) and viscous drag (i.e. dissipation) CANNOT be independent of each other. Rigorous proof of this is beyond our scope for now. We can however write the fluctuations in a 1-D LD equations as a source term.

46 Fluctuation-Dissipation Theorem The general form of a first order linear ODE is as follows: dy dx + p x y = q(x)

47 Fluctuation-Dissipation Theorem The general form of a first order linear ODE is as follows: dy dx + p x y = q(x) The solution to this equation is a linear combination of the general solution (i.e. q(x)=0) and the particular solution. y x = Ce x p t dt + e x p t dt x e s p t dt q s ds

48 Fluctuation-Dissipation Theorem The random-kick corrected equation of motion is of the same form, so making the appropriate substitutions and simplifying we have (PLEASE work through this at home): Τ v t = e ζt m v 0 + න 0 t Τ dt e ζ m t t δr(t )/m Now the correct ensemble mean squared velocity is given by: Square and take the average. < v 2 > = k BT m

49 Fluctuation-Dissipation Theorem < v 2 ζtτ >=< e m v 0 + න 0 t Τ dt e ζ m t t δr(t )/m 2 > 2ζt First Term: e m v Second Term: < δr t > = 0 Third Term: Τ t 0 dt e ] ζτm t t t δr(t )/m] ] 0 dt e ζτm t t δr(t )/m]

50 Fluctuation-Dissipation Theorem < v 2 ζtτ >=< e m v 0 + න 0 t Τ dt e ζ m t t δr(t )/m 2 > 2ζt First Term: e m v Second Term: < δr t > = 0 Third Term: Τ t 0 dt e ] ζτm t t t δr(t )/m] ] 0 dt e ζτm t t δr(t )/m] So, <v 2 >= 1 t m 2 0 e ζ(2t t t ) m < δr(t ) δr t Τ > dt dt

51 Fluctuation-Dissipation Theorem <v 2 >= 1 t m 2 0 e ζ(2t t t ) m < δr(t ) δr t Τ > dt dt Now, use Fubini s Theorem for solving the double integral and make a few variable transforms (I ll include a link at the end that details these transforms, they are a little involved but not difficult) Therefore, Τ <v 2 >= 1 t m 2 0 e ζ m < δr(0) δr t > dt

52 Fluctuation-Dissipation Theorem <v 2 >= 1 t m 2 0 e ζ(2t t t ) m < δr(t ) δr t Τ > dt dt Now, use Fubini s Theorem for solving the double integral and make a few variable transforms (I ll include a link at the end that details these transforms, they are a little involved but not difficult) Therefore, Τ <v 2 >= 1 t m 2 0 e ζ m < δr(0) δr t > dt = k B T/m

53 Fluctuation-Dissipation Theorem <v 2 >= 1 t m 2 0 e ζ(2t t t ) m < δr(t ) δr t Τ > dt dt Now, use Fubini s Theorem for solving the double integral and make a few variable transforms (I ll include a link at the end that details these transforms, they are a little involved but not difficult) Therefore, Τ <v 2 >= 1 t m 2 0 e ζ m < δr(0) δr t > dt = k B T/m Recall: <δr(t) δr(t )> = 2Bδ(t t ) (Autocorrelation Function)

54 Fluctuation-Dissipation Theorem <v 2 >= 1 t m 2 0 e ζ(2t t t ) m < δr(t ) δr t Τ > dt dt Now, use Fubini s Theorem for solving the double integral and make a few variable transforms (I ll include a link at the end that details these transforms, they are a little involved but not difficult) Therefore, Τ <v 2 >= 1 t m 2 0 e ζ m < δr(0) δr t > dt = k B T/m

55 Fluctuation-Dissipation Theorem Recall: <δr(t) δr(t )> = 2Bδ(t t ) (Autocorrelation Function) k So, we have B T = B 1 e 2ζtΤ m, m ζm which in the limit of t becomes B/ ζm Thus we have: <δr(t) δr(t )> = 2Bδ(t t )= 2k B Tζδ(t t )

56 Conclusions Through the course of the talk we have explored: Ensembles The Boltzmann Distribution Partition Functions Boltzmann s idea of Entropy The concept of thermostatting Velocity rescaling Langevin Dynamics And the dreaded link:

57 Conclusions Suggested reading: STRONGLY recommended (No, seriously.)

58 Conclusions Thank you.