Physics
Equipartition Theorem
The Equipartition Theorem states that in thermal equilibrium, each degree of freedom of a system will have an average energy of kT/2, where k is the Boltzmann constant and T is the temperature. This principle is used to predict the distribution of energy among different modes of motion in a system, such as translational, rotational, and vibrational motion of molecules.
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10 Key excerpts on "Equipartition Theorem"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902. Derivations Kinetic energies and the Maxwell–Boltzmann distribution The original formulation of the Equipartition Theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average kinetic energy, (3/2) k B T . This may be shown using the Maxwell–Boltzmann distribution, which is the probability distribution for the speed of a particle of mass m in the system, where the speed v is the magnitude of the velocity vector The Maxwell–Boltzmann distribution applies to any system composed of atoms, and assumes only a canonical ensemble, specifically, that the kinetic energies are distributed ________________________ WORLD TECHNOLOGIES ________________________ according to their Boltzmann factor at a temperature T . The average kinetic energy for a particle of mass m is then given by the integral formula as stated by the Equipartition Theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state. Quadratic energies and the partition function More generally, the Equipartition Theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax 2 , where A is a constant, has an average energy of ½ k B T in thermal equilibrium. In this case the Equipartition Theorem may be derived from the partition function Z ( β ), where β = 1/( k B T ) is the canonical inverse temperature. Integration over the variable x yields a factor in the formula for Z . The mean energy associated with this factor is given by as stated by the Equipartition Theorem. General proofs General derivations of the Equipartition Theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble and for the canonical ensemble.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902. Derivations Kinetic energies and the Maxwell–Boltzmann distribution The original formulation of the Equipartition Theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average kinetic energy, (3/2) k B T . This may be shown using the Maxwell–Boltzmann distribution, which is the probability distribution for the speed of a particle of mass m in the system, where the speed v is the magnitude of the velocity vector The Maxwell–Boltzmann distribution applies to any system composed of atoms, and assumes only a canonical ensemble, specifically, that the kinetic energies are distributed according to their Boltzmann factor at a temperature T . The average kinetic energy for a particle of mass m is then given by the integral formula ________________________ WORLD TECHNOLOGIES ________________________ as stated by the Equipartition Theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state. Quadratic energies and the partition function More generally, the Equipartition Theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax 2 , where A is a constant, has an average energy of ½ k B T in thermal equilibrium. In this case the Equipartition Theorem may be derived from the partition function Z ( β ), where β = 1/( k B T ) is the canonical inverse temperature. Integration over the variable x yields a factor in the formula for Z . The mean energy associated with this factor is given by as stated by the Equipartition Theorem. General proofs General derivations of the Equipartition Theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble and for the canonical ensemble.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
Here k B ¼ 1:381 10 23 J=K ¼ 8:617 10 5 eV=K is Boltzmann’s constant, T is the temperature in kelvin, and a degree of freedom refers to all independent quadratic terms in the energy. In thermal equilibrium, the internal energy of a system will be equally distributed over all accessible modes – we would not expect, for example, the energy in the x-direction motion to be significantly different than the energy in the y-direction motion. There are several important modifications of the Equipartition Theorem that are required by quantum theory, and we will discuss them as we encounter them. As an example of equipartition, consider a monatomic atom (helium, for example). The energy is given by just the kinetic energy E int ¼ KE ¼ 1 2 mv x 2 þ 1 2 mv y 2 þ 1 2 mv z 2 ¼ p x 2 2m þ p y 2 2m þ p z 2 2m : (12.1) Since the x-, y-, and z-velocities or momenta can vary independently, there are thus three independent quadratic terms, and the Equipartition Theorem predicts that the average energy (average kinetic energy) of each helium molecule is hEi ¼ 3 2 k B T . What are the implications of this prediction? First, it predicts that as the temperature decreases toward T ¼ 0 K the molecular kinetic energy goes to zero. Since we cannot have a negative kinetic energy, this implies that zero degrees on the Kelvin scale (273 C) is the lowest possible temperature, absolute zero. Second, this predicts the change in temperature of a system to which energy is added. If an amount of heat energy dQ added to a system results in a temperature change dT, the heat capacity of a system is defined as C ¼ dQ/dT. For a monatomic gas at constant volume, the Equipartition Theorem predicts the heat capacity should be C ¼ dQ dT ¼ dhE int i dT ¼ 3 2 k B per molecule. Heat capacity depends on the size of the system – the number of molecules. If the heat capacity is normalized by the system mass or number of particles, for example, this yields the material-dependent quantity specific heat.
- Carlo Di Castro, Roberto Raimondi(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
56 From Boltzmann to Boltzmann–Gibbs 3.5 The Equipartition Theorem One of the most important consequences of the Gibbs postulate is the Equipartition Theorem, telling how the energy of a system is divided, on average, among all the degrees of freedom of the system itself. Let us indicate with x i any of the 6 N coordinates upon which the Hamiltonian depends. By using the definition of the microcanonical average, we may write x i ∂ H ∂ x i = 1 ω( E ) d x i ∂ H ∂ x i δ( E − H ). (3.44) The integral can be transformed into a energy-constant surface integral x i ∂ H ∂ x i = 1 ω( E ) σ ( E ) x i ∂ H ∂ x i dσ ||∇ H || . (3.45) We now observe that ∇ H defines a vector orthogonal to the energy-constant surface and ∇ H /||∇ H || defines, correspondingly, the unit vector ˆ n normal to the surface. Then, by using the generalized divergence theorem, x i ∂ H ∂ x i = 1 ω( E ) σ ( E ) x i ˆ n i dσ = 1 ω( E ) H
- Martin Oliver Steinhauser(Author)
- 2012(Publication Date)
- De Gruyter(Publisher)
Section 3.3 Equilibrium distribution 253 This can be written as U=N D P j j exp .ˇ j / P j exp .ˇ j / : (3.87) The appropriate value of ˇ then is the one which, when put into this equation, pre-cisely produces the internal energy per particle .U=N/ , specified by the macrostate. Unfortunately, this is not a very tidy result, but it is as far as we can go explicitly, since in general, one cannot invert equation (3.87) to produce an explicit formula for ˇ as a function of .U;V;N/ . Nevertheless, for a given .U;V;N/ macrostate, ˇ is fully specified by (3.87), and one can indeed describe it as a “potential for energy”, in that the equation gives a very direct correlation between ( U=N ) and ˇ . It turns out that this untidy (but absolutely specific) function ˇ has a very clear physical meaning in terms of thermodynamic functions other than .U;V;N/ . In fact we shall see that it must be related to temperature only. This is a sufficiently important point to justify devoting the following section to it. 3.3.2 A statistical definition of temperature To show that there is a necessary relation between ˇ and the temperature T , we con-sider the thermodynamic and statistical treatment of two systems in thermal equilib-rium. The thermodynamic treatment is obvious. Two systems in thermal equilibrium have, effectively by definition, the same temperature. This statement is based on the zeroth law of thermodynamics , which states that there is some common function of state shared by all systems in mutual thermal equilibrium – and this function of state is what is meant by (empirical) temperature. The statistical treatment of T can follow directly, along the lines of Section 3.3. The problem can be set up as follows. Consider two systems A and B , which are in thermal contact with each other, but are together isolated from the rest of the universe. We suppose that system A consists of a fixed number N A of localized particles, which each have states with energy j , as before.
eBook - PDFTheoretical Concepts in Physics
An Alternative View of Theoretical Reasoning in Physics
- Malcolm S. Longair(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
To convert these ideas in a theory of statistical thermodynamics, we need to make some assumptions. Central to the subject is the principle of equal equilibrium probabilities. This is the statement that In statistical equilibrium, all quantum states accessible to the system are equally likely to be occupied. Thus, if we share three quanta among 100 oscillators, there are 171 700 ways of sharing out this energy and, in equilibrium, the system is equally likely to be found in any one of them. There is no point in disguising the fact that there are a number of problems with this assumption – for example, does the system really explore all possible distributions in the age of the Universe? Likewise, we need to assume that there are equal probabilities for a transition between any pair of states to occur in either direction. Waldram refers to this as the principle of jump-rate symmetry, which fortunately follows from the equality of the transition probabilities between quantum states according to quantum mechanics. These principles work incredibly well and so provide their own justification. This is the beginning of a journey which leads to the statistical definition of the law of increase of entropy and what we mean by temperature in statistical physics. Waldram gives a very pleasant example of how this works. Consider two large systems A and B, A containing N A = 5000 and B N B = 10 000 oscillators. All the oscillators have the same energy levels and so we can work out the density of states for each of them. For A, g A = ( N A + Q A − 1) ! Q A ! ( N A − 1) ! ≈ ( N A + Q A ) ! Q A ! N A ! , (12.71) where the last equality follows from the fact that the numbers are so large that the −1s are of no importance. We can now simplify this result using Stirling’s approximation in the form Q! ≈ Q Q . Therefore, g A = ( N A + Q A ) N A +Q A Q Q A A N N A A .
eBook - PDF- Wassim M. Haddad(Author)
- 2019(Publication Date)
- Princeton University Press(Publisher)
TEMPERATURE EQUIPARTITION AND THE KINETIC THEORY OF GASES 213 where α ij (0) = 0 and 0 < δ 1 ≤ δ 2 ; and ii ) the pairwise symmetry condition α ji ( ξ ) = -α ij ( -ξ ) . (4.33) The linear Fourier law, in which q ij = k ij ( T j -T i ) and q ji = k ij ( T i -T j ), is a heat transfer law of symmetric Fourier type. For the heat transfer law between two subsystems to be of symmetric Fourier type, the subsystems must be connected; the “heat transfer law” for an unconnected pair of subsystems, q ij = q ji = 0, violates the first inequality in the sector bound (4.32). If the heat transfer law between a pair of subsystems i and j is of symmetric Fourier type, then the energy transfer between those subsystems automatically satisfies the first and second laws of thermodynamics. To see that the first law of thermodynamics is satisfied, note that the time rate of change of the total internal energy of the subsystem pair is given by ˙ E ij + ˙ E ji = α ij ( T j -T i ) + α ji ( T i -T j ) = α ij ( T j -T i ) -α ij ( T j -T i ) = 0 , (4.34) where the second equality follows from the symmetry condition (4.33). To see that the second law of thermodynamics is satisfied, note that the time rate of change of the total entropy of the subsystem pair is given by ˙ S ij + ˙ S ji = α ij ( T j -T i ) T i + α ji ( T i -T j ) T j = 1 T i -1 T j α ij ( T j -T i ) = ( T j -T i ) T i T j α ij ( T j -T i ) ≥ 0 , (4.35) where the second equality follows from the symmetry condition (4.33) and the inequality follows from the sector bound condition (4.32). The sector bound condition also implies that equality holds in (4.35) if and only if T i = T j . If every pair of subsystems in a system is either disconnected or subject to a heat transfer law of symmetric Fourier type, then the first and second laws of thermodynamics will be satisfied at the system level. To see that the first law of thermodynamics is satisfied, consider the rate of change of
- Christopher R. Jacobs, Hayden Huang, Ronald Y. Kwon(Authors)
- 2012(Publication Date)
- Garland Science(Publisher)
Spring is in the air and a mosquito is in your apartment. You are watching it fly and land on a set of stairs. Assuming that it needs 1k B T of energy to fly up from one step to the next, what would a Boltzmann’s distribution predict for the probability of finding it on any given step, if there are 10 steps in total? 6. Consider a molecule that has a “home” position and a linear restoring force that is proportional to the distance that it is away from this position. In other words, the molecule acts like it is attached to the tip of a spring. The restoring force is F = −kx, where x is the distance from the home position and k is the spring constant. The potential energy is a function of position and can be expressed as W k x = 1 2 2 . ANNOTATED REFERENCES 149 We can treat each molecular position of the molecule, x, as a microstate of the system. In this case, assuming thermal equilibrium, the probability of finding the molecule in any given position is given by the Boltzmann distribution. What is the average energy? Hint: a table of integral identities may be useful. Note: your finding is known as the principle of equipartition of energy and holds for the energy associated with any parameter that has an associated energy that varies with the square of the parameter. For example, it also works for kinetic energy, which varies with the square of speed. 7. In this problem, we show that for an isolated system, equilibrium is attained by maximizing the entropy (this is the second law of thermodynamics and is equivalent to stating that equilibrium is attained by minimizing the free energy, assuming constant energy). Imagine a box that is isolated from the external environment, meaning that the total energy of the contents within the box is constant. The box contains a partition that divides it into two subspaces.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
Also, the system one considers in those cases is a single particle state , not a particle. (In the above discussion, we could have assumed our system to be a single atom.) Limits of applicability The Bose–Einstein and Fermi–Dirac distributions may be written: ________________________ WORLD TECHNOLOGIES ________________________ Assuming the minimum value of ε i is small, it can be seen that the condition under which the Maxwell–Boltzmann distribution is valid is when For an ideal gas, we can calculate the chemical potential using the development in the Sackur–Tetrode article to show that: where E is the total internal energy, S is the entropy, V is the volume, and Λ is the thermal de Broglie wavelength. The condition for the applicability of the Maxwell– Boltzmann distribution for an ideal gas is again shown to be 3. Partition function (statistical mechanics) In statistical mechanics, the partition function , Z , is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its deri-vatives. There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.
eBook - PDF- Myron Kaufman(Author)
- 2002(Publication Date)
- CRC Press(Publisher)
5 Statistical Mechanics Perhaps, the atomic hypothesis will be replaced by another some day— perhaps, but not probably. Ludwig Boltzmann In this chapter, the macroscopic world of thermodynamics is linked to the microscopic world of atoms and molecules through the idea of the probability of a macroscopic state. The basic assumption of statistical mechanics is that this probability is proportional to the number of microscopic arrangements that can give the macroscopic state. This number is calculated for distributions in space and among energy levels. The Boltzmann distribution is derived from the second law of thermodynamics. The partition function is used to obtain formulas to calculate thermodynamic functions from molecular properties. The development is applied to a system with only translational energy to give the partition function and thermodynamic properties of the monatomic ideal gas. The Maxwell velocity distribution and several of its averages from kinetic theory are calculated. Extension of the theory to polyatomic molecules in the classical limit gives the classical Equipartition Theorem. Foreshadowing the discussion of steady-state systems, thermal transpiration is treated by kinetic theory. Statistical theory is Copyright © 2002 by Taylor & Francis Group LLC applied to the model of the freely jointed chain to discuss the mechanical properties of macromolecules. 5.1 The Microscopic World Our discussion of thermodynamics has been thoroughly grounded on observa- tions on the real macroscopic world. It is not necessary to be aware of the existence of atoms in order to discuss and use the first, second, and third laws of thermodynamics. The basis for the interpretation of thermodynamics in terms of the microscopic world of atoms, molecules, and ions was first given by Ludwig Boltzmann in 1896.
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