where (E) is the partition function of the uniform ensemble. Recalling that we have (E) = E (E) (E) i = ij x (E) j E = ij ln (E) E = k ij ~ S E = kt i

Size: px
Start display at page:

Download "where (E) is the partition function of the uniform ensemble. Recalling that we have (E) = E (E) (E) i = ij x (E) j E = ij ln (E) E = k ij ~ S E = kt i"

Transcription

1 G25.265: Statistical Mechanics Notes for Lecture 4 I. THE CLASSICAL VIRIAL THEOREM (MICROCANONICAL DERIVATION) Consider a system with Hamiltonian H(x). Let x i and x j be specic components of the phase space vector. The classical virial theorem states that i = kt ij where the average is taken with respect to a microcanonical ensemble. To prove the theorem, start with the denition of the average: i = C (E) dxx i (E? H(x)) where the fact that (x) = (?x) has been used. Also, the N and V dependence of the partition function have been suppressed. Note that the above average can be written as However, writing allows the average to be expressed as i = i = C (E) E = C (E) = C (E) E E dx x i (E? H(x)) dx x i dx x i (H? E) x i (H? E) = [x i (H? E)]? ij (H? E) C (E) E = C (E) E I H=E dx [x i (H? E)] + ij (E? H(x)) x i (H? E)dS j + ij dx (E? H(x)) The rst integral in the brackets is obtained by integrating the total derivative with respect to x j over the phase space variable x j. This leaves an integral that must be performed over all other variables at the boundary of phase space where H = E, as indicated by the surface element ds j. But the integrand involves the factor H? E, so this integral will vanish. This leaves: i = C ij (E) = C ij (E) E = ij (E) (E) dx H<E dx(e? H(x))

2 where (E) is the partition function of the uniform ensemble. Recalling that we have (E) = E (E) (E) i = ij x (E) j E = ij ln (E) E = k ij ~ S E = kt ij which proves the theorem. Example: x i = p i and i = j. The virial theorem says that hp i p i i = kt h p2 i m i i = kt h p2 i 2m i i = 2 kt Thus, at equilibrium, the kinetic energy of each particle must be kt=2. By summing both sides over all the particles, we obtain a well know result 3NX i= h p2 i 2m i i = 3NX i= h 2 m iv 2 i i = 3 2 NkT II. LEGENDRE TRANSFORMS The microcanonical ensemble involved the thermodynamic variables N, V and E as its variables. However, it is often convenient and desirable to work with other thermodynamic variables as the control variables. Legendre transforms provide a means by which one can determine how the energy functions for dierent sets of thermodynamic variables are related. The general theory is given below for functions of a single variable. Consider a function f(x) and its derivative y = f 0 (x) = df dx g(x) The equation y = g(x) denes a variable transformation from x to y. Is there a unique description of the function f(x) in terms of the variable y? That is, does there exist a function (y) that is equivalent to f(x)? Given a point x 0, can one determine the value of the function f(x 0 ) given only f 0 (x 0 )? No, for the reason that the function f(x 0 ) + c for any constant c will have the same value of f 0 (x 0 ) as shown in the gure below. 2

3 f(x) x 0 x b(x ) 0 FIG.. However, the value f(x 0 ) can be determined uniquely if we specify the slope of the line tangent to f at x 0, i.e., f 0 (x 0 ) and the y-intercept, b(x 0 ) of this line. Then, using the equation for the line, we have This relation must hold for any general x: f(x 0 ) = x 0 f 0 (x 0 ) + b(x 0 ) f(x) = xf 0 (x) + b(x) Note that f 0 (x) is the variable y, and x = g? (y), where g? is the functional inverse of g, i.e., g(g? (x)) = x. Solving for b(x) = b(g? (y)) gives b(g? (y)) = f(g? (y))? yg? (y) (y) where (y) is known as the Legendre transform of f(x). In shorthand notation, one writes (y) = f(x)? xy however, it must be kept in mind that x is a function of y. III. THE CANONICAL ENSEMBLE A. Basic Thermodynamics In the microcanonical ensemble, the entropy S is a natural function of N,V and E, i.e., S = S(N; V; E). This can be inverted to give the energy as a function of N,V, and S, i.e., E = E(N; V; S). Consider using Legendre transformation to change from S to T using the fact that 3

4 E T = S The Legendre transform ~ E of E(N; V; S) is ~E(N; V; T ) = E(N; V; S(T ))? S E S = E(N; V; S(T ))? T S The quantity ~ E(N; V; T ) is called the Hemlholtz free energy and is given the symbol A(N; V; T ). It is the fundamental energy in the canonical ensemble. The dierential of A is A da = T However, from A = E? T S, we have From the rst law, de is given by Thus, A dt + V N;T da = de? T ds? SdT A dv + N de = T ds? P dv + dn da =?P dv? SdT + dn Comparing the two expressions, we see that the thermodynamic relations are A S =? T A P =? V N;T A = N V;T T;V dn B. The partition function Consider two systems ( and 2) in thermal contact such that N 2 N E 2 E N = N + N 2 ; E = E + E 2 dim(x ) dim(x 2 ) and the total Hamiltonian is just H(x) = H (x ) + H 2 (x 2 ) Since system 2 is innitely large compared to system, it acts as an innite heat reservoir that keeps system at a constant temperature T without gaining or losing an appreciable amount of heat, itself. Thus, system is maintained at canonical conditions, N; V; T. The full partition function (N; V; E) for the combined system is the microcanonical partition function (N; V; E) = dx(h(x)? E) = dx dx 2 (H (x ) + H 2 (x 2 )? E) Now, we dene the distribution function, f(x ) of the phase space variables of system as 4

5 f(x ) = dx 2 (H (x ) + H 2 (x 2 )? E) Taking the natural log of both sides, we have ln f(x ) = ln dx 2 (H (x ) + H 2 (x 2 )? E) Since E 2 E, it follows that H 2 (x 2 ) H (x ), and we may expand the above expression about H = 0. To linear order, the expression becomes (x ) ln dx 2 (H (x ) + H 2 (x 2 )? E) ln f(x ) = ln dx 2 (H 2 (x 2 )? E) + H (x ) = ln dx 2 (H 2 (x 2 )? E)? H (x ) E ln dx 2 (H 2 (x 2 )? E) H (x)=0 where, in the last line, the dierentiation with respect to H is replaced by dierentiation with respect to E. Note that ln E ln dx 2 (H 2 ( 2 )? E) = S 2(E) k dx 2 (H 2 (x 2 )? E) = S 2 (E) E k = kt where T is the common temperature of the two systems. Using these two facts, we obtain ln f(x ) = S 2(E) k? H (x ) kt f(x ) = e S2(E)=k e?h(x)=kt Thus, the distribution function of the canonical ensemble is f(x) / e?h(x)=kt The prefactor exp(s 2 (E)=k) is an irrelevant constant that can be disregarded as it will not aect any physical properties. The normalization of the distribution function is the integral: dxe?h(x)=kt Q(N; V; T ) where Q(N; V; T ) is the canonical partition function. It is convenient to dene an inverse temperature = =kt. Q(N; V; T ) is the canonical partition function. As in the microcanonical case, we add in the ad hoc quantum corrections to the classical result to give Q(N; V; T ) = N!h 3N dxe?h(x) The thermodynamic relations are thus, Hemlholtz free energy: A(N; V; T ) =? ln Q(N; V; T ) To see that this must be the denition of A(N; V; T ), recall the denition of A: A = E? T S = hh(x)i? T S 5

6 But we saw that A S =? T Substituting this in gives A = hh(x)i? T A T or, noting that A T = A T =? A kt 2 it follows that A = hh(x)i + A This is a simple dierential equation that can be solved for A. We will show that the solution is A =? ln Q() Note that Substituting in gives, therefore so this form of A satises the dierential equation. Other thermodynamics follow: A = ln Q()? Q Q = A? hh(x)i A = hh(x)i + A? hh(x)i = A Average energy: E = hh(x)i = Q C N dxh(x)e?h(x) =? ln Q(N; V; T ) Pressure: A P =? V N;T = kt ln Q(N; V; T ) V N;T Entropy: S =? A T =? A T = A kt 2 = k 2? ln Q(N; V; T ) =?k ln Q + k ln Q = ke + k ln Q = k ln Q + E T 6

7 Heat capacity at constant volume: E C V = T = E T = k2 ln Q(N; V; T ) 2 C. Relation between canonical and microcanonical ensembles We saw that the E(N; V; S) and A(N; V; T ) could be related by a Legendre transformation. The partition functions (N; V; E) and Q(N; V; T ) can be related by a Laplace transform. Recall that the Laplace transform f() ~ of a function f(x) is given by ~f() = 0 dxe?x f(x) Let us compute the Laplace transform of (N; V; E) with respect to E: ~(N; V; ) = C N dee?e dx(h(x)? E) Using the -function to do the integral over E: ~(N; V; ) = C N 0 dxe?h(x) By identifying =, we see that the Laplace transform of the microcanonical partition function gives the canonical partition function Q(N; V; T ). D. Classical Virial Theorem (canonical ensemble derivation) Again, let x i and x j canonical average given by But Thus, be specic components of the phase space vector x = (p ; :::; p 3N ; q ; :::; q 3N ). Consider the x i i = Q C N = Q C N e?h(x) = i =? Q C N =? Q C N = i dxx i e?h(x) x j dxx i? e?h(x) x i e?h(x)? e?h(x) x i = x i e?h(x)? ij e?h(x) dx x i e?h(x) + Q ijc N dx 0 dx 0 x i e?h(x) dx j x i e?h(x) + kt ij x j =? + kt ij dxe?h(x) Several cases exist for the surface term x i exp(?h(x)): 7

8 . x i = p i a momentum variable. Then, since H p 2 i, exp(?h) evaluated at p i = clearly vanishes. 2. x i = q i and U! as q i!, thus representing a bound system. Then, exp(?h) also vanishes at q i =. 3. x i = q i and U! 0 as q i!, representing an unbound system. Then the exponential tends to both at q i =, hence the surface term vanishes. 4. x i = q i and the system is periodic, as in a solid. Then, the system will be represented by some supercell to which periodic boundary conditions can be applied, and the coordinates will take on the same value at the boundaries. Thus, H and exp(?h) will take on the same value at the boundaries and the surface term will vanish. 5. x i = q i, and the particles experience elastic collisions with the walls of the container. Then there is an innite potential at the walls so that U?! at the boundary and exp(?h)?! 0 at the bondary. Thus, we have the result i = kt ij The above cases cover many but not all situations, in particular, the case of a system conned within a volume V with reecting boundaries. Then, surface contributions actually give rise to an observable pressure (to be discussed in more detail in the next lecture). 8

Thus, the volume element remains the same as required. With this transformation, the amiltonian becomes = p i m i + U(r 1 ; :::; r N ) = and the canon

Thus, the volume element remains the same as required. With this transformation, the amiltonian becomes = p i m i + U(r 1 ; :::; r N ) = and the canon G5.651: Statistical Mechanics Notes for Lecture 5 From the classical virial theorem I. TEMPERATURE AND PRESSURE ESTIMATORS hx i x j i = kt ij we arrived at the equipartition theorem: * + p i = m i NkT

More information

G : Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into

G : Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into G25.2651: Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into thermal contact. By thermal contact, we mean that the

More information

[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B

[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B Canonical ensemble (Two derivations) Determine the probability that a system S in contact with a reservoir 1 R to be in one particular microstate s with energy ɛ s. (If there is degeneracy we are picking

More information

in order to insure that the Liouville equation for f(?; t) is still valid. These equations of motion will give rise to a distribution function f(?; t)

in order to insure that the Liouville equation for f(?; t) is still valid. These equations of motion will give rise to a distribution function f(?; t) G25.2651: Statistical Mechanics Notes for Lecture 21 Consider Hamilton's equations in the form I. CLASSICAL LINEAR RESPONSE THEORY _q i = @H @p i _p i =? @H @q i We noted early in the course that an ensemble

More information

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity

More information

Preliminary Examination: Thermodynamics and Statistical Mechanics Department of Physics and Astronomy University of New Mexico Fall 2010

Preliminary Examination: Thermodynamics and Statistical Mechanics Department of Physics and Astronomy University of New Mexico Fall 2010 Preliminary Examination: Thermodynamics and Statistical Mechanics Department of Physics and Astronomy University of New Mexico Fall 2010 Instructions: The exam consists of 10 short-answer problems (10

More information

Lecture 6: Ideal gas ensembles

Lecture 6: Ideal gas ensembles Introduction Lecture 6: Ideal gas ensembles A simple, instructive and practical application of the equilibrium ensemble formalisms of the previous lecture concerns an ideal gas. Such a physical system

More information

Grand Canonical Formalism

Grand Canonical Formalism Grand Canonical Formalism Grand Canonical Ensebmle For the gases of ideal Bosons and Fermions each single-particle mode behaves almost like an independent subsystem, with the only reservation that the

More information

Part II Statistical Physics

Part II Statistical Physics Part II Statistical Physics Theorems Based on lectures by H. S. Reall Notes taken by Dexter Chua Lent 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

A Brief Introduction to Statistical Mechanics

A Brief Introduction to Statistical Mechanics A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade

More information

Nanoscale simulation lectures Statistical Mechanics

Nanoscale simulation lectures Statistical Mechanics Nanoscale simulation lectures 2008 Lectures: Thursdays 4 to 6 PM Course contents: - Thermodynamics and statistical mechanics - Structure and scattering - Mean-field approaches - Inhomogeneous systems -

More information

(i) T, p, N Gibbs free energy G (ii) T, p, µ no thermodynamic potential, since T, p, µ are not independent of each other (iii) S, p, N Enthalpy H

(i) T, p, N Gibbs free energy G (ii) T, p, µ no thermodynamic potential, since T, p, µ are not independent of each other (iii) S, p, N Enthalpy H Solutions exam 2 roblem 1 a Which of those quantities defines a thermodynamic potential Why? 2 points i T, p, N Gibbs free energy G ii T, p, µ no thermodynamic potential, since T, p, µ are not independent

More information

Introduction Statistical Thermodynamics. Monday, January 6, 14

Introduction Statistical Thermodynamics. Monday, January 6, 14 Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can

More information

Statistical Mechanics

Statistical Mechanics Statistical Mechanics Newton's laws in principle tell us how anything works But in a system with many particles, the actual computations can become complicated. We will therefore be happy to get some 'average'

More information

Lecture 2. Derivative. 1 / 26

Lecture 2. Derivative. 1 / 26 Lecture 2. Derivative. 1 / 26 Basic Concepts Suppose we wish to nd the rate at which a given function f (x) is changing with respect to x when x = c. The simplest idea is to nd the average rate of change

More information

New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space

New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Lesson 6: Linear independence, matrix column space and null space New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Two linear systems:

More information

Classical Statistical Mechanics: Part 1

Classical Statistical Mechanics: Part 1 Classical Statistical Mechanics: Part 1 January 16, 2013 Classical Mechanics 1-Dimensional system with 1 particle of mass m Newton s equations of motion for position x(t) and momentum p(t): ẋ(t) dx p =

More information

9.1 System in contact with a heat reservoir

9.1 System in contact with a heat reservoir Chapter 9 Canonical ensemble 9. System in contact with a heat reservoir We consider a small system A characterized by E, V and N in thermal interaction with a heat reservoir A 2 characterized by E 2, V

More information

PHYS 352 Homework 2 Solutions

PHYS 352 Homework 2 Solutions PHYS 352 Homework 2 Solutions Aaron Mowitz (, 2, and 3) and Nachi Stern (4 and 5) Problem The purpose of doing a Legendre transform is to change a function of one or more variables into a function of variables

More information

G : Statistical Mechanics

G : Statistical Mechanics G25.2651: Statistical Mechanics Notes for Lecture 15 Consider Hamilton s equations in the form I. CLASSICAL LINEAR RESPONSE THEORY q i = H p i ṗ i = H q i We noted early in the course that an ensemble

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability

More information

5.62 Physical Chemistry II Spring 2008

5.62 Physical Chemistry II Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.62 Lecture #9: CALCULATION

More information

1 The fundamental equation of equilibrium statistical mechanics. 3 General overview on the method of ensembles 10

1 The fundamental equation of equilibrium statistical mechanics. 3 General overview on the method of ensembles 10 Contents EQUILIBRIUM STATISTICAL MECHANICS 1 The fundamental equation of equilibrium statistical mechanics 2 2 Conjugate representations 6 3 General overview on the method of ensembles 10 4 The relation

More information

2. Thermodynamics. Introduction. Understanding Molecular Simulation

2. Thermodynamics. Introduction. Understanding Molecular Simulation 2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular

More information

Gases and the Virial Expansion

Gases and the Virial Expansion Gases and the irial Expansion February 7, 3 First task is to examine what ensemble theory tells us about simple systems via the thermodynamic connection Calculate thermodynamic quantities: average energy,

More information

Introduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables!

Introduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables! Introduction Thermodynamics: phenomenological description of equilibrium bulk properties of matter in terms of only a few state variables and thermodynamical laws. Statistical physics: microscopic foundation

More information

Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.

Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density

More information

Dually Flat Geometries in the State Space of Statistical Models

Dually Flat Geometries in the State Space of Statistical Models 1/ 12 Dually Flat Geometries in the State Space of Statistical Models Jan Naudts Universiteit Antwerpen ECEA, November 2016 J. Naudts, Dually Flat Geometries in the State Space of Statistical Models. In

More information

U U B U P x

U U B U P x Smooth Quantum Hydrodynamic Model Simulation of the Resonant Tunneling Diode Carl L. Gardner and Christian Ringhofer y Department of Mathematics Arizona State University Tempe, AZ 8587-184 Abstract Smooth

More information

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 1 Reading: 3.1-3.5 Chandler, Chapters 1 and 2 McQuarrie This course builds on the elementary concepts of statistical

More information

Chapter 10: Statistical Thermodynamics

Chapter 10: Statistical Thermodynamics Chapter 10: Statistical Thermodynamics Chapter 10: Statistical Thermodynamics...125 10.1 The Statistics of Equilibrium States...125 10.2 Liouville's Theorem...127 10.3 The Equilibrium Distribution Function...130

More information

Statistical Mechanics. Atomistic view of Materials

Statistical Mechanics. Atomistic view of Materials Statistical Mechanics Atomistic view of Materials What is statistical mechanics? Microscopic (atoms, electrons, etc.) Statistical mechanics Macroscopic (Thermodynamics) Sample with constrains Fixed thermodynamics

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Friday, January 17, 2014 1:00PM to 3:00PM General Physics (Part I) Section 5. Two hours are permitted for the completion of this section

More information

8.044 Lecture Notes Chapter 8: Chemical Potential

8.044 Lecture Notes Chapter 8: Chemical Potential 8.044 Lecture Notes Chapter 8: Chemical Potential Lecturer: McGreevy Reading: Baierlein, Chapter 7. So far, the number of particles N has always been fixed. We suppose now that it can vary, and we want

More information

( ) Lectures on Hydrodynamics ( ) =Λ Λ = T x T R TR. Jean-Yve. We can always diagonalize point by point. by a Lorentz transformation

( ) Lectures on Hydrodynamics ( ) =Λ Λ = T x T R TR. Jean-Yve. We can always diagonalize point by point. by a Lorentz transformation Lectures on Hydrodynamics Jean-Yve µν T µ ( x) 0 We can always diagonalize point by point by a Lorentz transformation T Λ µν x ( u ) µ and space-rotation R ( Ω) Then, in general, λ0 0 0 0 0 λx 0 0 Λ Λ

More information

IV. Classical Statistical Mechanics

IV. Classical Statistical Mechanics IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed

More information

The Partition Function Statistical Thermodynamics. NC State University

The Partition Function Statistical Thermodynamics. NC State University Chemistry 431 Lecture 4 The Partition Function Statistical Thermodynamics NC State University Molecular Partition Functions In general, g j is the degeneracy, ε j is the energy: = j q g e βε j We assume

More information

although Boltzmann used W instead of Ω for the number of available states.

although Boltzmann used W instead of Ω for the number of available states. Lecture #13 1 Lecture 13 Obectives: 1. Ensembles: Be able to list the characteristics of the following: (a) icrocanonical (b) Canonical (c) Grand Canonical 2. Be able to use Lagrange s method of undetermined

More information

STATISTICAL PHYSICS II

STATISTICAL PHYSICS II . STATISTICAL PHYSICS II (Statistical Physics of Non-Equilibrium Systems) Lecture notes 2009 V.S. Shumeiko 1 . Part I CLASSICAL KINETIC THEORY 2 1 Lecture Outline of equilibrium statistical physics 1.1

More information

2.2 The derivative as a Function

2.2 The derivative as a Function 2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)

More information

Energy Barriers and Rates - Transition State Theory for Physicists

Energy Barriers and Rates - Transition State Theory for Physicists Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle

More information

Introduction into thermodynamics

Introduction into thermodynamics Introduction into thermodynamics Solid-state thermodynamics, J. Majzlan Chemical thermodynamics deals with reactions between substances and species. Mechanical thermodynamics, on the other hand, works

More information

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

More information

i=1 n i, the canonical probabilities of the micro-states [ βǫ i=1 e βǫn 1 n 1 =0 +Nk B T Nǫ 1 + e ǫ/(k BT), (IV.75) E = F + TS =

i=1 n i, the canonical probabilities of the micro-states [ βǫ i=1 e βǫn 1 n 1 =0 +Nk B T Nǫ 1 + e ǫ/(k BT), (IV.75) E = F + TS = IV.G Examples The two examples of sections (IV.C and (IV.D are now reexamined in the canonical ensemble. 1. Two level systems: The impurities are described by a macro-state M (T,. Subject to the Hamiltonian

More information

LECTURE 15 + C+F. = A 11 x 1x1 +2A 12 x 1x2 + A 22 x 2x2 + B 1 x 1 + B 2 x 2. xi y 2 = ~y 2 (x 1 ;x 2 ) x 2 = ~x 2 (y 1 ;y 2 1

LECTURE 15 + C+F. = A 11 x 1x1 +2A 12 x 1x2 + A 22 x 2x2 + B 1 x 1 + B 2 x 2. xi y 2 = ~y 2 (x 1 ;x 2 ) x 2 = ~x 2 (y 1 ;y 2  1 LECTURE 5 Characteristics and the Classication of Second Order Linear PDEs Let us now consider the case of a general second order linear PDE in two variables; (5.) where (5.) 0 P i;j A ij xix j + P i,

More information

Physics 4230 Final Examination 10 May 2007

Physics 4230 Final Examination 10 May 2007 Physics 43 Final Examination May 7 In each problem, be sure to give the reasoning for your answer and define any variables you create. If you use a general formula, state that formula clearly before manipulating

More information

Thermodynamics. Basic concepts. Thermal equilibrium and temperature

Thermodynamics. Basic concepts. Thermal equilibrium and temperature hermodynamics Basic concepts hermodynamics is a phenomenological description of macroscopic systems with many degrees of freedom. Its microscopic justication is provided by statistical mechanics. Equilibrium

More information

ChE 503 A. Z. Panagiotopoulos 1

ChE 503 A. Z. Panagiotopoulos 1 ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,

More information

I.G Approach to Equilibrium and Thermodynamic Potentials

I.G Approach to Equilibrium and Thermodynamic Potentials I.G Approach to Equilibrium and Thermodynamic otentials Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For eample, in the previous section we

More information

CHEM-UA 652: Thermodynamics and Kinetics

CHEM-UA 652: Thermodynamics and Kinetics 1 CHEM-UA 652: Thermodynamics and Kinetics Notes for Lecture 4 I. THE ISOTHERMAL-ISOBARIC ENSEMBLE The isothermal-isobaric ensemble is the closest mimic to the conditions under which most experiments are

More information

Chemistry. Lecture 10 Maxwell Relations. NC State University

Chemistry. Lecture 10 Maxwell Relations. NC State University Chemistry Lecture 10 Maxwell Relations NC State University Thermodynamic state functions expressed in differential form We have seen that the internal energy is conserved and depends on mechanical (dw)

More information

MA 102 Mathematics II Lecture Feb, 2015

MA 102 Mathematics II Lecture Feb, 2015 MA 102 Mathematics II Lecture 1 20 Feb, 2015 Differential Equations An equation containing derivatives is called a differential equation. The origin of differential equations Many of the laws of nature

More information

Ensemble equivalence for non-extensive thermostatistics

Ensemble equivalence for non-extensive thermostatistics Physica A 305 (2002) 52 57 www.elsevier.com/locate/physa Ensemble equivalence for non-extensive thermostatistics Raul Toral a;, Rafael Salazar b a Instituto Mediterraneo de Estudios Avanzados (IMEDEA),

More information

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF OSLO FCULTY OF MTHEMTICS ND NTURL SCIENCES Exam in: FYS430, Statistical Mechanics Day of exam: Jun.6. 203 Problem :. The relative fluctuations in an extensive quantity, like the energy, depends

More information

P 1 P * 1 T P * 1 T 1 T * 1. s 1 P 1

P 1 P * 1 T P * 1 T 1 T * 1. s 1 P 1 ME 131B Fluid Mechanics Solutions to Week Three Problem Session: Isentropic Flow II (1/26/98) 1. From an energy view point, (a) a nozzle is a device that converts static enthalpy into kinetic energy. (b)

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #10

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #10 MASSACHUSES INSIUE OF ECHNOLOGY Physics Department 8.044 Statistical Physics I Spring erm 203 Problem : wo Identical Particles Solutions to Problem Set #0 a) Fermions:,, 0 > ɛ 2 0 state, 0, > ɛ 3 0,, >

More information

2m + U( q i), (IV.26) i=1

2m + U( q i), (IV.26) i=1 I.D The Ideal Gas As discussed in chapter II, micro-states of a gas of N particles correspond to points { p i, q i }, in the 6N-dimensional phase space. Ignoring the potential energy of interactions, the

More information

G : Statistical Mechanics

G : Statistical Mechanics G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically

More information

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON UNIVERSITY OF SOUTHAMPTON PHYS2024W1 SEMESTER 2 EXAMINATION 2011/12 Quantum Physics of Matter Duration: 120 MINS VERY IMPORTANT NOTE Section A answers MUST BE in a separate blue answer book. If any blue

More information

Chapter 4: Partial differentiation

Chapter 4: Partial differentiation Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of

More information

quantum semiconductor devices. The model is valid to all orders of h and to rst order in the classical potential energy. The smooth QHD equations have

quantum semiconductor devices. The model is valid to all orders of h and to rst order in the classical potential energy. The smooth QHD equations have Numerical Simulation of the Smooth Quantum Hydrodynamic Model for Semiconductor Devices Carl L. Gardner and Christian Ringhofer y Department of Mathematics Arizona State University Tempe, AZ 8587-84 Abstract

More information

PHYS 705: Classical Mechanics. Hamiltonian Formulation & Canonical Transformation

PHYS 705: Classical Mechanics. Hamiltonian Formulation & Canonical Transformation 1 PHYS 705: Classical Mechanics Hamiltonian Formulation & Canonical Transformation Legendre Transform Let consider the simple case with ust a real value function: F x F x expresses a relationship between

More information

Some properties of the Helmholtz free energy

Some properties of the Helmholtz free energy Some properties of the Helmholtz free energy Energy slope is T U(S, ) From the properties of U vs S, it is clear that the Helmholtz free energy is always algebraically less than the internal energy U.

More information

Our goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always

Our goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.

More information

Statistical Mechanics in a Nutshell

Statistical Mechanics in a Nutshell Chapter 2 Statistical Mechanics in a Nutshell Adapted from: Understanding Molecular Simulation Daan Frenkel and Berend Smit Academic Press (2001) pp. 9-22 11 2.1 Introduction In this course, we will treat

More information

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook

More information

Entrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems

Entrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion.

More information

I.G Approach to Equilibrium and Thermodynamic Potentials

I.G Approach to Equilibrium and Thermodynamic Potentials I.G Approach to Equilibrium and Thermodynamic Potentials Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For eample, in the previous section we

More information

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics Reading/Homework Assignment Read chapter 3 in Rose. Midterm Exam, April 5 (take home)

More information

[A + 1 ] + (1 ) v: : (b) Show: the derivative of T at v = v 0 < 0 is: = (v 0 ) (1 ) ; [A + 1 ]

[A + 1 ] + (1 ) v: : (b) Show: the derivative of T at v = v 0 < 0 is: = (v 0 ) (1 ) ; [A + 1 ] Homework #2 Economics 4- Due Wednesday, October 5 Christiano. This question is designed to illustrate Blackwell's Theorem, Theorem 3.3 on page 54 of S-L. That theorem represents a set of conditions that

More information

Legendre-Fenchel transforms in a nutshell

Legendre-Fenchel transforms in a nutshell 1 2 3 Legendre-Fenchel transforms in a nutshell Hugo Touchette School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK Started: July 11, 2005; last compiled: August 14, 2007

More information

U = γkt N. is indeed an exact one form. Thus there exists a function U(S, V, N) so that. du = T ds pdv. U S = T, U V = p

U = γkt N. is indeed an exact one form. Thus there exists a function U(S, V, N) so that. du = T ds pdv. U S = T, U V = p A short introduction to statistical mechanics Thermodynamics consists of the application of two principles, the first law which says that heat is energy hence the law of conservation of energy must apply

More information

3.320 Lecture 18 (4/12/05)

3.320 Lecture 18 (4/12/05) 3.320 Lecture 18 (4/12/05) Monte Carlo Simulation II and free energies Figure by MIT OCW. General Statistical Mechanics References D. Chandler, Introduction to Modern Statistical Mechanics D.A. McQuarrie,

More information

summary of statistical physics

summary of statistical physics summary of statistical physics Matthias Pospiech University of Hannover, Germany Contents 1 Probability moments definitions 3 2 bases of thermodynamics 4 2.1 I. law of thermodynamics..........................

More information

University of Illinois at Chicago Department of Physics SOLUTIONS. Thermodynamics and Statistical Mechanics Qualifying Examination

University of Illinois at Chicago Department of Physics SOLUTIONS. Thermodynamics and Statistical Mechanics Qualifying Examination University of Illinois at Chicago Department of Physics SOLUTIONS Thermodynamics and Statistical Mechanics Qualifying Eamination January 7, 2 9: AM to 2: Noon Full credit can be achieved from completely

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =

More information

RATIONAL POINTS ON CURVES. Contents

RATIONAL POINTS ON CURVES. Contents RATIONAL POINTS ON CURVES BLANCA VIÑA PATIÑO Contents 1. Introduction 1 2. Algebraic Curves 2 3. Genus 0 3 4. Genus 1 7 4.1. Group of E(Q) 7 4.2. Mordell-Weil Theorem 8 5. Genus 2 10 6. Uniformity of Rational

More information

Lectures 15: Parallel Transport. Table of contents

Lectures 15: Parallel Transport. Table of contents Lectures 15: Parallel Transport Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this lecture we study the

More information

Statistical mechanics of classical systems

Statistical mechanics of classical systems Statistical mechanics of classical systems States and ensembles A microstate of a statistical system is specied by the complete information about the states of all microscopic degrees of freedom of the

More information

The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit:

The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k B T µ, βµ 1, which defines the degenerate Fermi gas. In this

More information

First Law of Thermodyamics U = Q + W. We can rewrite this by introducing two physical. Enthalpy, H, is the quantity U + pv, as this

First Law of Thermodyamics U = Q + W. We can rewrite this by introducing two physical. Enthalpy, H, is the quantity U + pv, as this First Law of Thermodyamics U = Q + W where U is the increase in internal energy of the system, Q is the heat supplied to the system and W is the work done on the system. We can rewrite this by introducing

More information

The Second Virial Coefficient & van der Waals Equation

The Second Virial Coefficient & van der Waals Equation V.C The Second Virial Coefficient & van der Waals Equation Let us study the second virial coefficient B, for a typical gas using eq.v.33). As discussed before, the two-body potential is characterized by

More information

Diffusion equation, flux, diffusion coefficient scaling. Diffusion in fully ionized plasma vs. weakly ionized plasma. n => Coulomb collision frequency

Diffusion equation, flux, diffusion coefficient scaling. Diffusion in fully ionized plasma vs. weakly ionized plasma. n => Coulomb collision frequency Last Time Diffusion in plasma: the main reason why we need to control it (i.e. using magnetic field) Diffusion equation, flux, diffusion coefficient scaling o o t nx,t Dn D2 nx,t o D ~ L 2 T Diffusion

More information

The Microcanonical Approach. (a) The volume of accessible phase space for a given total energy is proportional to. dq 1 dq 2 dq N dp 1 dp 2 dp N,

The Microcanonical Approach. (a) The volume of accessible phase space for a given total energy is proportional to. dq 1 dq 2 dq N dp 1 dp 2 dp N, 8333: Statistical Mechanics I Problem Set # 6 Solutions Fall 003 Classical Harmonic Oscillators: The Microcanonical Approach a The volume of accessible phase space for a given total energy is proportional

More information

Thermodynamics at Small Scales. Signe Kjelstrup, Sondre Kvalvåg Schnell, Jean-Marc Simon, Thijs Vlugt, Dick Bedeaux

Thermodynamics at Small Scales. Signe Kjelstrup, Sondre Kvalvåg Schnell, Jean-Marc Simon, Thijs Vlugt, Dick Bedeaux Thermodynamics at Small Scales Signe Kjelstrup, Sondre Kvalvåg Schnell, Jean-Marc Simon, Thijs Vlugt, Dick Bedeaux At small scales: 1. The systematic method of Hill 2. Small systems: surface energies 3.

More information

3.1 Derivative Formulas for Powers and Polynomials

3.1 Derivative Formulas for Powers and Polynomials 3.1 Derivative Formulas for Powers and Polynomials First, recall that a derivative is a function. We worked very hard in 2.2 to interpret the derivative of a function visually. We made the link, in Ex.

More information

Physics 408 Final Exam

Physics 408 Final Exam Physics 408 Final Exam Name You are graded on your work, with partial credit where it is deserved. Please give clear, well-organized solutions. 1. Consider the coexistence curve separating two different

More information

Math 121, Practice Questions for Final (Hints/Answers)

Math 121, Practice Questions for Final (Hints/Answers) Math 11, Practice Questions for Final Hints/Answers) 1. The graphs of the inverse functions are obtained by reflecting the graph of the original function over the line y = x. In each graph, the original

More information

We already came across a form of indistinguishably in the canonical partition function: V N Q =

We already came across a form of indistinguishably in the canonical partition function: V N Q = Bosons en fermions Indistinguishability We already came across a form of indistinguishably in the canonical partition function: for distinguishable particles Q = Λ 3N βe p r, r 2,..., r N ))dτ dτ 2...

More information

Dynamics that trigger/inhibit cluster formation in a one-dimensional granular gas

Dynamics that trigger/inhibit cluster formation in a one-dimensional granular gas Physica A 342 (24) 62 68 www.elsevier.com/locate/physa Dynamics that trigger/inhibit cluster formation in a one-dimensional granular gas Jose Miguel Pasini a;1, Patricio Cordero b; ;2 a Department of Theoretical

More information

2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?

2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x? Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f

More information

The fine-grained Gibbs entropy

The fine-grained Gibbs entropy Chapter 12 The fine-grained Gibbs entropy 12.1 Introduction and definition The standard counterpart of thermodynamic entropy within Gibbsian SM is the socalled fine-grained entropy, or Gibbs entropy. It

More information

Brief Review of Statistical Mechanics

Brief Review of Statistical Mechanics Brief Review of Statistical Mechanics Introduction Statistical mechanics: a branch of physics which studies macroscopic systems from a microscopic or molecular point of view (McQuarrie,1976) Also see (Hill,1986;

More information

B. Physical Observables Physical observables are represented by linear, hermitian operators that act on the vectors of the Hilbert space. If A is such

B. Physical Observables Physical observables are represented by linear, hermitian operators that act on the vectors of the Hilbert space. If A is such G25.2651: Statistical Mechanics Notes for Lecture 12 I. THE FUNDAMENTAL POSTULATES OF QUANTUM MECHANICS The fundamental postulates of quantum mechanics concern the following questions: 1. How is the physical

More information

a factors The exponential 0 is a special case. If b is any nonzero real number, then

a factors The exponential 0 is a special case. If b is any nonzero real number, then 0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the

More information

1 The pendulum equation

1 The pendulum equation Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating

More information

Section 6.1: Composite Functions

Section 6.1: Composite Functions Section 6.1: Composite Functions Def: Given two function f and g, the composite function, which we denote by f g and read as f composed with g, is defined by (f g)(x) = f(g(x)). In other words, the function

More information

Elementary ODE Review

Elementary ODE Review Elementary ODE Review First Order ODEs First Order Equations Ordinary differential equations of the fm y F(x, y) () are called first der dinary differential equations. There are a variety of techniques

More information

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order

More information

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.

More information