Statistical Mechanics for Engineers

It is in this context that the method of statistical mechanics becomes important to us. Instead, the emphasis is on making the basic ideas of statistical mechanics accessible to the intended audience.

Classical Mechanics

Inertial Frame

In fact, the coordinate system attached to the desk, in this case, is no longer an inertial frame until the train resumes its motion along a straight line at a constant speed.

Mechanics of a Single Particle

Newton’s Equation of Motion
Work
Potential Energy
Kinetic Energy
Conservation of Energy

Now let's calculate the work required to move the particle an infinitesimal distance from rtor+dr. 1.26). The resulting expression for Wi is seen to depend only on the mass of the particle and its velocities att1andt2.

Fig. 1.1 The sum of infinitesimal displacements dr i results in a net displacement dr = ∑ n i=1 dr i =

Mechanics of Many Particles

Mechanical Energy of a Many-Particle System

Let us calculate the work required to produce a certain change in the mechanical state of a many-particle system. N (1.44) gives the force exerted on the last particle by all the other particles in the system.

Center of Mass

The first two terms of (1.56) are the mechanical energy the collection of N particles would have if all the particles were concentrated at R to form a single material point of mass M. The remaining term is the mechanical energy of the N particles calculated in a (generally non-inertial) coordinate system whose origin is attached to R.

Hamilton’s Principle

Lagrangian and Action
Generalized Coordinates
Many Mechanical Degrees of Freedom

The configuration of the system is uniquely determined if we specify the position of the particles (x1,y1) and (x2,y2). You can assume that the motion of the spring and particle is limited to

Fig. 1.6 The actual path z(t) followed by the mechanical system (thick curve) and a varied path z ′ (t) = z(t) +δ z(t) (thin wavy curve)

Momentum and Energy: Definitions

Alternatively, you can prove the claim through a more straightforward calculation of the various derivatives, as illustrated in the next exercise. As seen in the example below, it is often equal to mechanical energy, that is, the sum of kinetic and potential energies.

1.7 † Energy Function and Energy

Conservation Laws and Symmetry

Conservation of Linear Momentum
Conservation of Energy

This is a consequence of the isotropy of space, that is, all directions in space are equivalent. So the constants of motion are the energy and the linear momentum component.

Hamilton’s Equations of Motion

So the energy, the x-andy components of linear momentum, and the thez-component of angular momentum are constants of motion. If the field source is bounded tox>0, the system is invariant only with respect to (1) translation inside and (2) translation along their axis.

1.10 † Routhian

Poisson Bracket

Thus, a dynamic variable can be a constant of motion even if it has an explicit time dependence. This is known as Poisson's theorem, which allows us on occasion to generate a new constant of motion from known constants of motion.

Frequently Used Symbols

Thermodynamics

The First Law of Thermodynamics

However, our everyday experience tells us that a change in the apparent state of the system can be brought about without doing any work on it. We use the symbol W to denote the work in this narrower sense of the word.

Quantifying Heat

Suppose that an external body exerts a force dA on the system through a surface element of area dA on the boundary of the system. When this quantity is added together for all such surface elements, the end result is the net change in the volume of the system.

The Second Law of Thermodynamics

Equilibrium of an Isolated System

Thus, we expect that the state of the considered system is not in equilibrium. According to the second law, processes that cause a decrease in entropy are impossible for an isolated system.

Fundamental Equations

Closed System of Fixed Composition
Open System
Heat Capacities

Equilibrium condition 1 For an isolated system to be in equilibrium, it is necessary and sufficient that the entropy of the system is (locally) maximal under a set of constraints imposed on the system. Thus, the work done on a system by an infinitesimal displacement of an infinitesimal part of the boundary of the system depends on the position of this surface element. This means that (2.22) and (2.36) are equivalent ways of expressing the same set of information about the thermodynamic behavior of the system.

Fig. 2.2 The function S = S(U), for some fixed values of V , N 1 , . . . , N c , can be inverted to give U = U(S) since 1/T = (∂ S/∂ U) V,N > 0, and hence the function is monotonic as indicated in a.

Role of Additional Variables

Thus, the last term (2.76) can be interpreted as the heat introduced into the system by the flow of molecules. the last term indicates the flow of entropy associated with the flow of molecules. To make the role of additional variables clearer, consider the example shown in Figure 2.4. The interior of the box is divided into two parts by a partition, the properties of which can be changed if necessary.

Fig. 2.4 A system consisting of two compartments. The system as a whole is isolated, while the properties of the wall separating the compartments can be modified as needed.

Entropy Representation

Condition of Equilibrium
Equality of Temperature
Direction of a Spontaneous Process

You can assume that the temperatures of the two compartments are equal at all times. If =s(u) is concave up as shown in Fig.2.6, the system is unstable with respect to splitting into two homogeneous parts with distinct values of internal energy density. Then, the entropy density of the system in this final state is given by sf in the figure.

Fig. 2.5 Various possible states of equilibrium when X 1 is confined to the interval X 1 a ,X 1 b

Energy Representation

Reversible Work Source

The vertical arrow U0 indicates the trajectory of the system during evolution towards equilibrium as observed in the U S plane. Let us first plot the equilibrium value of S of the system as a function of its U as shown in Fig.2.10. If the system is allowed to relax to equilibrium in the absence of the external agency, then as in Fig.

Fig. 2.7 Graphical representation of S of the composite system considered in Sect. 2.8.2

Euler Relation

Note that the expansion of the system boundary does not affect the physical state of matter anywhere in the homogeneous body. Now let us consider the procedure in which the partition is enlarged to contain a larger part of the homogeneous body. Again, this is a mental process and has no effect on the physical state of the body, including the values of u,s and n1,.

Fig. 2.12 A system taken inside a homogeneous body is indicated by the smaller circle

Gibbs–Duhem Relation

2.12 ‡ Gibbs Phase Rule

Free Energies

Fixing Temperature
Direction of a Spontaneous Process
Fundamental Equation
Other Free Energies

Now, assume that the composite system consisting of the heat bath and the system of interest, as a whole, is isolated. If the composite system is in equilibrium, then the system is obviously in equilibrium at the temperature T of the heat bath. Conversely, if the system is in equilibrium atT, the composite system is also in equilibrium because of the assumptions made about the heat bath.

Fig. 2.13 The system of interest in contact with a heat bath. The reversible work source is con- con-nected only to the system and not to the heat bath.

2.14 ‡ Maxwell Relation

2.15 ‡ Partial Molar Quantities

Graphical Methods

Pure Systems: F Versus V (Constant T )

Gibbs–Duhem Relation at Constant T : Interpretation

Binary Mixtures: G Versus x 1 (Constant T and P)

Phase Separation
Phase Coexistence

If we draw a tangent at a certain value of V, say V∗, the minus slope is the pressure P∗at(T,V∗). The negative value of the slope and the intersection on the F-axis of the second tangent are the pressure and chemical potential at (T,V∗+dV), denoted by P∗+∆P and µ∗+∆ µ, respectively. Due to the way the diagram is drawn, the Gversusx1plot is concave down at xa1 and xb1.

Fig. 2.15 Graphical construction involving an F versus V isotherm.

Frequently Used Symbols

His definition of an adiabatic process differs from ours in that he requires that the process is also reversible. Nishioka K (1987) An analysis of the Gibbs theory of infinitesimal discontinuous variation in interface thermodynamics. Whitaker S (1992) Introduction to Fluid Mechanics, Krieger Publishing Company, Florida Very readable and carefully written introduction to fluid mechanics.

Classical Statistical Mechanics

Macroscopic Measurement
Phase Space
Ensemble Average
Statistical Equilibrium
Statistical Ensemble
Liouville’s Theorem
Significance of H

The characteristic time scale for this type of molecular motion is of the order of 10−12s. At a given instant t, each copy of the ensemble has a representative point, as specified by (qf(t),pf(t)), in phase space. This implies that the number density Nρ of the copies satisfies the equivalent of mass balance or the continuity equation from fluid mechanics in phase space.

Fig. 3.1 Measurement of the length of a bar.

3.8 † The Number of Constants of Motion

Canonical Ensemble

Each particle is subject to the external field generated by the walls of the cylinder. 2m +φ(rN) +ψ(rN,λ), (3.55) whereφ is the potential energy due to mutual interaction between the particles, andψ is the net external field generated by the piston and the other walls of the cylinder. Instead, it is a parameter that characterizes the distribution of the members of the statistical ensemble over states with different energy values.

Fig. 3.4 Gas particles confined to a cylinder. Each particle is subject to the external field generated by the walls of the cylinder

Simple Applications of Canonical Ensemble

Rectangular Coordinate System
Equipartition Theorem
Spherical Coordinate System

However, the numerical values of these integrals should be independent of the choice of coordinate system we use. It should be emphasized that the Maxwell–Boltzmann distribution holds regardless of the shape. By performing integrations with respect to pφ, pθ and pr using the formula from exercise 3.4, we find

Fig. 3.5 A simple model of a polar molecule.

Canonical Ensemble and Thermal Contact

Furthermore, suppose that the interaction between A and B is also weak in the sense just defined. It is significant that the additivity of the free energy relies on the fact that the interaction between the two subsystems A and B is quite weak. But, we remember that the interaction between the system and the surroundings was supposed to be quite weak.

Corrections from Quantum Mechanics

A System of Identical Particles
Implication of the Uncertainty Principle
Applicability of Classical Statistical Mechanics

Att=0, we can mentally label all the molecules in the system and measure their positions and velocities. As we saw in the previous section, if there are identical particles in the system, and the integration over qfandpfindesPdistinct permutations of identical particles, we have. Nevertheless, we find it convenient to interpret as "the sum of states" as before and consider.

Fig. 3.6 In classical mechanics, a and b represent two distinct microstates (r 1 ,r 2 ,p 1 ,p 2 ) = (r,r ′ ,p,p ′ ) and (r 1 ,r 2 ,p 1 ,p 2 ) = (r ′ ,r,p ′ ,p), respectively

3.13 † A Remark on the Statistical Approach

These observations suggest that the proper construction of the probability density ρ is not quite what we saw in Section 3.3. To facilitate the calculation of the partial derivative, we assume that the system is a cube of side lengthL=V1/3 and introduce a new set of variables from. 3.185) is the negative of the net force exerted on its particle by the other particles in the system.

3.15 † Internal Energy

We will therefore first have to find the Hamiltonian H of the system suitable for O. 3.240) These equations indicate that the effect of the external field vanishes in a coordinate system moving with the center of mass of the system. The macroscopic motion of the center of mass accounts for the remaining three degrees of freedom.

3.16 † Equilibrium of an Accelerating Body

Frequently Used Symbols

From (3.271) we see that the Jacobian of the coordinate transformation from (rN,pN) to (rN,πN) is unity. Our treatment of a rotating body is based on their treatment of the subject, especially sects. A derivation of the continuity equation from this more satisfactory point of view can be found in Chap.

Various Statistical Ensembles

Fluctuations in a Canonical Ensemble

If we let H=1 J for the sake of illustration, then the magnitude of the fluctuation ∆rmsHis is of the order of 10−12J. Such an extremely small quantity cannot be measured by an instrument whose intended range is of the order of 1 J. We divide the body into M equal parts and indicate the value of the extensive quantity belonging to that part of the part.

Microcanonical Ensemble

Expression for ρ
Choice of ∆ E
Isolated System

Now we remember that copies in the original canonical ensemble are distributed in the phase space according to the density ρ which depends only on the energy of the system. Thus, the copies in the ensemble are distributed with uniform density in the region of phase space corresponding to (E−∆E,E). Note that the integral in (4.18) is the volume of the region in phase space (or abbreviated the phase volume) that is compatible with E−∆E

Phase Integral in Microcanonical Ensemble

Assume that in the initial state one of the compartments is filled with N non-interacting identical particles, while the other compartment is empty. Insofar as this process is irreversible, we expect that the entropy of the system in the final state is greater than in the initial state. The change in entropy after removing the partition is therefore given by Sf−Si=kBlnCMf −kBlnCMi =kBlnCMf.

Fig. 4.1 Step function. If H > E, E − H < 0 and hence θ(E − H) = 0. If H ≤ E, E − H ≥ 0 and θ(E − H) = 1.

4.4 † Adiabatic Reversible Processes

Canonical Ensemble

Closed System Held at a Constant Temperature
Canonical Distribution

Thus, in order to calculate p(Ea)dEa, we need to determine the total number of microstates AB that are consistent with (4.58) and the condition. The desired probability is this number divided by the total number of AB microstates, which is simply Ωab(Eab)∆E. In order to find the total number of microstates accessible to AB, we first find the number of microstates accessible to AB when Has is in a certain interval denoted by (4.59).