The “Arrow” of Time

In statistical thermodynamics we return often to the same statement of the equilibrium condition

Systems will spontaneously tend towards configurations of increasing number of microstates. The equilibrium state is the state of maximum microstate number -

the end point of this process.

This statement about the direction of spontaneous change has deep implications. It is the “rule” that defines the forward direction of time. As we observe a system undergoing a change, the direction of change is towards the configuration with the most microstates, not the reverse …. in the limit of large N and E.

Flash Quiz!

What are 5 ways you might be able to tell if a movie clip was running forward or backward?

Do they all share something in common?

Answer? What are 5 ways you might be able to tell if a movie clip was running forward or backward?

Do they all share something in common?

The final configuration has more microstates than the initial one… ?

http://www.youtube.com/watch?v=cKoZw97OiVk Suddenly getting up from total disorder into perfect running stride http://www.youtube.com/watch?v=tcrkN9v15JA Disordered heat/sound turned into concentrated gravitational energyhttp://www.youtube.com/watch?v=WmgDOUPegeIhttp://www.youtube.com/watch?v=N7PoLf2pib0http://www.youtube.com/watch?v=jMLwXJyB2K4 Ordered crystals growing from a disordered solution??

Let’s Consider the Dynamics of this Approach to Equilibrium

Dynamics of this process can be thought of as a series of random events in which particles either move from left to right or from right to left. Time enters when we assign a time for each individual event.

Later we shall look at how to predict the average time required per particle move.

The Dog-Flea Experiment

originally called the Ehrenfests’ Urn Model

1 2 3 2R2R-12R-2

Rover Rex

• Start with all 2R fleas on Rover.

• Every second, randomly pick a number between 1 and 2R and have that flea jump to the other dog.

• If NRover = the number of fleas on Rover, plot

NRover/R as a function of time.

Dynamics of a Random Process

The probability P+(NRover) of a flea jumping from Rover to Rex is just P+

(NRover) = NRover/2R

and the probability of the flea jumping from Rex to Rover is

P-(NRover) = 1-P+(NRover)

Rover Rex

At equilibrium the probability of a flea jumping off Rover must equal the probability of one jumping onto Rover – i.e. NRover = R at equilibrium

You can treat this system as one with 2R particles (the fleas) and two degenerate energy levels (Rover and Rex).

The configuration with the maximum number for microstates is that for which NRover=NRex=R

Dynamics of a Random Process

Rover Rex

0.5

1

1.5

2

0 1 2 3 4

No. of time steps/R

N-R

ov

er/

R R=20

R=80

R=200

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R R=20

R=80

R=200

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R=80

R=200

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R=80

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R=200NR

over/R

Experiment 1

Experiment 6

Experiment 2

Experiment 3

Experiment 4

Experiment 5

No. of steps/R

No. of steps/R

No. of steps/R

• The larger the number of fleas (i.e. 2R) the smaller the random fluctuations in NRover

• Each experiment differs in the nature of the random fluctuations

• NRover, averaged over the different experiments, decays to R, the equilibrium value

NR

over/R

Fluctuations Occur Even at Equilibrium

Start with both dogs having the same number of fleas and the random jumps of fleas between dogs produces fluctuations in the number of fleas on Rover (and Rex).

The average size of fluctuations in NRover is, like the average value of NRover, a property of the equilibrium state that has important physical consequences.

0

1

0 2 4 6 8 10 12 14 16

NRover/R

No. of steps/R

Instead of dogs and fleas ...

‘Dogs’ ‘Fleas’ Phenomenon Kinetic Coefficient

Two Samples Quanta of Energy Heat flow Thermal conductivity

Two regions within a liquid

Solute Molecules Solute diffusion Diffusion constant

Two different chemical species

Molecular transformations

Reversible unimolecular reaction

Chemical reaction rate coefficient

e.g.

the unimolecular reaction

initial later

time

In a laboratory setting we cannot count microstates, so it would be convenient to identify a macroscopic quantity that plays the same predictive role for spontaneous change.

When we examined the problem of heat flow, we established a relationship between the macroscopic quantity heat flow and the microscopic change in number of microstates.

Can we identify the direction of change without counting microstates?

Entropy and Spontaneous Change

Heat flow and microstate number are related by

Using the definition of temperature, we can re-write this as

This allows (or even inspires) us to define a quantity called entropy as

The entropy change

…predicts the change in number of microstates, and the direction of spontaneous change in terms of heat flow and temperature – macroscopic observables. From now on, we are able to dispense with microscopic considerations, and develop classical thermodynamics that is concerned only with the relationship between macroscopic (system) properties like T, P, E, heat…

The equilibrium condition can now be written as dS = 0.The necessary condition for a spontaneous process is that dS > 0.

ln heatd W dE

ln heatB

dEk d W

T

lnBS k W

ln heatB

dEdS k d W

T

Thermodynamics

We have replaced a quantity that depends on lots of microscopic information, i.e.

number of microstates of the equilibrium configuration

by a new quantity

entropy

that can be defined in terms of macroscopic heat flows. Now we can apply all our conditions for equilibrium and spontaneous change (derived statistically) in all the many situations which are too complicated to treat using statistical thermodynamics.

Understanding the consequences of these conditions – now called the Laws of Thermodynamics – is the goal of the next set of lectures

Entropy and Condition for Equilibrium

From statistical thermodynamics we defined entropy as

S = kBlnW

and defined the equilibrium state for an isolated system as that which maximised W, the number of microstates.

The problem is that we can’t usually count microstates.

So, we’ll restate the equilibrium condition as follows –

The equilibrium state of an isolated system (i.e. one characterised by a fixed energy U, volume V and particle number N) is that which maximises the entropy S. For any process in a such a system to occur spontaneously, it must involve a change in S, S 0.

Entropy without counting microstates

What do we know about entropy?

1. It depends only on U the energy, N the number of particles and V the volume.

In shorthand, S = S(U,V,N)

2. Entropy change is proportional to heat flow q

dS = q/T

For this relation to hold, the heat flow has to be slow enough so that the system moved incrementally from equilibrium state to equilibrium state. Such a heat flow is called quasi-static.

(Not Another) Definition of Temperature

If we have kept N and V constant, then the heat flow into the system produces an increase in energy i.e. dU = q

That means dS = dU/T for a constant N and V

This relation between a change in U and a change in S can be written most neatly in terms of partial derivatives, i.e.

This means we can define temperature with microscopic energy distributions by

TU

S

NV

1

,

NVS

UT

,

Extensive and Intensive Properties

S, U and V are called extensive properties because they are proportional to N

The temperature T doesn’t depend on N and is called an intensive property.

Here are two more important intensive properties.

Pressure

Chemical potential

While we can measure T and p, we generally can’t measure μ directly.

NSV

UP

,

VSN

U

,

Summary

You should now be able to• Define entropy statistically and macroscopically, and explain

the relationship between entropy, spontaneous change, and equilibrium.

Next Lecture

• Changes in Energy – Heat and Work