Lecture 4. Macrostates and Microstates (Ch. 2 )

Lecture 4. Macrostates and Microstates (Ch. 2 )

The past three lectures: we have learned about thermal energy, how it is stored at the microscopic level, and how it can be transferred from one system to another. However, the energy conservation law (the first law of thermodynamics) tells us nothing about the directionality of processes and cannot explain why so many macroscopic processes are irreversible. Indeed, according to the 1st law, all processes that conserve energy are legitimate, and the reversed-in-time process would also conserve energy. Thus, we still cannot answer the basic question of thermodynamics: why does the energy spontaneously flow from the hot object to the cold object and never the other way around? (in other words, why does the time arrow exist for macroscopic processes?).

For the next three lectures, we will address this central problem using the ideas of statistical mechanics. Statistical mechanics is a bridge from microscopic states to averages. In brief, the answer will be: irreversible processes are not inevitable, they are just overwhelmingly probable. This path will bring us to the concept of entropy and the second law of thermodynamics.

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Microstates and Macrostates

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2

The evolution of a system can be represented by a trajectory in the multidimensional (configuration, phase) space of micro-

parameters. Each point in this space represents a microstate.

During its evolution, the system will only pass through accessible microstates

? the ones that do not violate the conservation laws: e.g., for an isolated

system, the total internal energy must be conserved.

Microstate: the state of a system specified by describing the quantum state

of each molecule in the system. For a classical particle ? 6 parameters (xi, yi, zi, pxi, pyi, pzi), for a macro system ? 6N parameters.

The statistical approach: to connect the macroscopic observables

(averages) to the probability for a certain microstate to appear along the system's trajectory in configuration space, P( 1, 2,..., N).

Macrostate: the state of a macro system specified by its macroscopic

parameters. Two systems with the same values of macroscopic parameters are thermodynamically indistinguishable. A macrostate tells us nothing about a state of an individual particle.

For a given set of constraints (conservation laws), a system can be in many macrostates.

The Phase Space vs. the Space of Macroparameters

some macrostate P

T

V

the surface

defined by an

equation of

i

states

1 i

numerous microstates in a multi-dimensional configuration (phase) space that correspond the same macrostate

i

2

1

i

2 etc., etc., etc. ...

2

2

1

1

Examples: Two-Dimensional Configuration Space

motion of a particle in a one-dimensional box

-L

L

K=K0

0

K

"Macrostates" are characterized by a single parameter: the kinetic energy K0

Another example: one-dimensional

px

harmonic oscillator

U(r) K + U =const

x

px

-L

Lx

-px

Each "macrostate" corresponds to a continuum of x microstates, which are characterized by specifying the

position and momentum

The Fundamental Assumption of Statistical Mechanics

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The ergodic hypothesis: an isolated system in an

equilibrium state, evolving in time, will pass through all the

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2

accessible microstates at the same recurrence rate, i.e. all accessible microstates are equally probable.

microstates which correspond to the same energy

The ensemble of all equi-energetic states a mirocanonical ensemble.

Note that the assumption that a system is isolated is important. If a system is coupled to a heat reservoir and is able to exchange energy, in order to replace the system's trajectory by an ensemble, we must determine the relative occurrence of states with different energies. For example, an ensemble whose states' recurrence rate is given by their Boltzmann factor (e-E/kBT) is called a canonical ensemble.

The average over long times will equal the average over the ensemble of all equienergetic microstates: if we take a snapshot of a system with N microstates, we will find the system in any of these microstates with the same probability.

Probability for a stationary system

many identical measurements on a single system

a single measurement on many copies of the system

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