Micro-canonical ensemble model of particles obeying Bose-Einstein and Fermi-Dirac statistics

Size: px
Start display at page:

Download "Micro-canonical ensemble model of particles obeying Bose-Einstein and Fermi-Dirac statistics"

Transcription

1 Indian Journal o Pure & Applied Physics Vol. 4, October 004, pp Micro-canonical ensemble model o particles obeying Bose-Einstein and Fermi-Dirac statistics Y K Ayodo, K M Khanna & T W Sakwa Department o Physical Sciences, Western University College, Box 90, Kakamega, Kenya Department o Physics, Moi University, Box 5 Eldoret, Kenya Received 3 February 004; accepted April 004 A micro-canonical ensemble or an assembly o bosons and ermions is considered in which the number o particles, internal energy and volume are kept constant. A statistical distribution model, which is ermion dominated and where bosons and ermions interact in pairs, is developed. The partition unction is derived. Macroscopic thermodynamic quantities such as entropy, internal energy and speciic heat are obtained in terms o the partition unction. The model equations are applied to a mixture o liquid helium-3 and liquid helium-4 atoms. [Keywords: Bose-Einstein Statistics, Fermi-Dirac statistics, Partition unction, 3 He- 4 He mixture] IPC Code: C0B 3/00 Introduction A micro-canonical ensemble represents a collection o conigurations o isolated systems that have reached thermal equilibrium. A system is isolated rom its environment i it does not exchange either particles or energy with its surroundings. The volume, internal energy and the number o particles o such a system are constant and are the same or all conigurations that are part o the same microcanonical ensemble. In this paper, a coniguration o a mixture o bosons and ermions is studied and a partition unction is developed or the same. Thermodynamic quantities, such as internal energy, speciic heat and entropy can be calculated rom the knowledge o the statistical distribution and the partition unction. So ar most o the studies deal either with a system o bosons or with a system o ermions. In nature, there do exist systems, which are mixtures o bosons and ermions such as H, H and 3 H, and the most interesting mixture is 4 He and 3 He. It should be clearly understood that in the mixture, bosons obey Bose-Einstein statistics and ermions obey Fermi-Dirac statistics. What distribution law or what will be the expression or the most probable distribution-in-energy in the mixture is the subect matter o study in this paper. The irst attempt to generalise quantum Bose and Fermi statistics or a mixture o Bosons and Fermions was made by Gentile. He proposed statistics in which up to N particles (N>> were allowed to occupy a single quantum state instead o ust one particle or Fermi case due to the Pauli exclusion principle, and ininitely many or the Bose case. However, Gentile s approach was ound to be too much o a generalisation and contained the violation o the conventionally accepted Pauli principle. Furthermore, his model did not distinguish which particles were ermions and which ones were bosons. However, Gentile s work laid the emphasis and the oundation that the statistical mechanics o a mixture o bosons and ermions can be worked out. The next attempt was that o Medvedev. In his paper entitled properties o particles obeying ambiguous statistics, Medvedev proposed a new class o identical particles, which may exhibit both Bose and Fermi statistics with respective probabilities P b and P. The model admits only primary Bose- Einstein and Fermi-Dirac statistics as existing. He assumed that a particle is neither a pure boson nor a pure ermion. He let another particle, which interacts with the irst one, play the role o an external observer. During the interaction it perorms a measurement at the irst particle and identiies it as either a boson or a ermion with respective probabilities P b and P. According to the result o this measurement, it interacts with the irst particle as i the last is a ermion or a boson, respectively. The

2 750 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 irst particle is the observer or the second particle and so the process is symmetric. Note that (P b + P is not necessarily equal to one, and, i not, it means that the second particle (observer does not detect the irst particle. The probability o this is ( P b P. The statistical uncertainty introduced here may be either the intrinsic property o a particle itsel or the experimental uncertainty o the measurement process. Another attempt in this direction is the so-called statistical independence model o Landau and Lishitz 3 in which two weakly interacting subsystems (bosons and ermions are together regarded as one composite system, and the subsystems are assumed to be quasi-closed. The statistical distribution or count or such a mixture is the product o the individual probabilities or two subsystems, one corresponding to bosons and the other corresponding to ermions. With these assumptions, the statistical independence model will hold only or an ideal gas assembly o bosons and ermions. In reality, such an assembly does not exist and hence the statistical independence model cannot be used or real mixtures o bosons and ermions like 3 He and 4 He mixtures. Chan et al. 4 studied the eect o disorder on superluid 3 He- 4 He mixtures, and the thermodynamics o 3 He- 4 He mixtures in aerogel. However, the studies related to more o an ideal system rather than a real system. In our earlier paper 5 entitled statistical mechanics and thermodynamics or a mixture o bosons and ermions, the statistical distribution model or a mixture that was dominated by bosons was developed. The partition unction that was derived worked well or a liquid helium-3 and liquid helium-4 mixture. In this paper, an assembly that is ermion dominated is studied. Thereore, the properties o a mixture o bosons and ermions assuming there exists a pair interaction between the bosons and ermions; and that the concentrations o the bosons and ermions are dierent rom each other are studied. Since the concentrations o the bosons and ermions will not be the same, and considering only pair interaction, in a given state o equilibrium some ermions will be let unpaired. The value o the occupation number o ermions in a given state will not exceed, rather, will be much less, than the degeneracy o that state so that Pauli exclusion principle is not violated. With these basic assumptions, the expressions or the ollowing were derived: (i Statistical count or an ensemble that is a mixture o the bosons and ermions assuming a pair interaction between the bosons and ermions. (ii The most probable distribution in energy or a mixture o the bosons and the ermions in the ensemble. (iii The partition unction or such an ensemble. (iv Using the partition unction, the calculations were done or internal energy, speciic heat and entropy. The above expressions are used to study the thermodynamic properties o a mixture o liquid helium-3 (ermions and liquid helium-4 (bosons with dierent concentrations. For our model calculations a ermion concentration o 0.70 was used. The theoretical results obtained are compared with the experimental observations on the properties o a mixture o liquid helium-3 and liquid helium-4. Theory Consider a micro-canonical assembly o N particles in which there are N b bosons and N ermions such that N= N b + N ( Let ε, ε, ε 3 ε be the energy states o the assembly, and in the statistical equilibrium the number o particles assigned to these energy levels be n, n, n 3, n, respectively, such that the numbers n must satisy the conditions requiring the conservation o particles, N, and conservation o energy, E, i.e., n = N ( = and = n ε = E (3 such that, n = nb + n (4 where, n b = number o bosons in the energy level (5

3 AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 75 and n =number o ermions in the energy level (6 P b = n ( b ω ( ω ( nb ( ω n ( n nb (9 Let ω be the number o states in the -th level, i.e, ω is the degeneracy o the -th level. Then the number o ways P b in which n b bosons can be assigned to ω states in the -th level is given by, P b n ( ω b = (7 Similarly, the number o ways, P, in which n ermions can be assigned to the ω states is given by, P = ω ( ω n (8 To satisy Pauli exclusion principle, it is necessary to assume that ω >> n. Once the particles are placed in the ω sub-levels in the -th level, we shall urther assume that bosons and ermions may interact in pairs. However, not all the ermions may orm pairs with bosons since the bosons and the ermions will not be in equal proportions in the mixture. This statement implies that n > n b. It is the number o bosons n b that will determine the number o boson-ermion pairs. Hence, the number o boson-ermion pairs will be n b and the number o unpaired ermions will be (n n b. Since the permutations among the particles and the permutations among the pairs in the same energy level do not give a new complexion, in the statistical distribution model proposed here, the ollowing permutations must be excluded rom the number o ways in which n b bosons, n ermions and n b bosonermion pairs are distributed in the -th level: (i Permutations among identical pairs should be excluded by dividing by (n b (ii Permutations among identical unpaired ermions should be excluded by dividing by (n n b Hence the total number o ways, P b, in which n b bosons, n ermions and n b pairs o bosons and ermions can be distributed among the ω sub-levels in the -th level is given by, The statistical count, C b, or such a distribution among all the levels ( =,,3 available to the assembly is the product o such expressions as given in Eq. (9 since every arrangement in a given energy level can be considered independently o the other energy levels. Thus we can write, C b = Pb = nb ( ω ( ω ( n ( ω n ( n n = = b b (0 Equation (0 will now be used to calculate the most probable distribution in energy or the ensemble. 3 Most probable distribution in energy or the ensemble The obective is to calculate or what values o n b and n, the statistical count C b, is maximum under the conditions o N and E being ixed. The distribution numbers and the corresponding energies must satisy the ollowing relations: n = N ( = = b b n ε = E ( = b b n = N (3 = n ε = E (4 where N b and N are the total number o bosons and ermions in the ensemble such that the total number o particles N is given by, N= N b + N (5 Similarly, E b and E are the total internal energies o bosons and ermions such that the total energy o the ensemble E is given by,

4 75 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 E= E b + E (6 Now to ind the values o n b and n or which C b is maximum, the procedure is to allow the variation o C b with respect to n b and n, and put the result equal to zero. For C b to be maximum, ln( Cb = ln( Cb dnb = nb + ln( Cb dn = 0 = n (7 The variations dn b and dn are not independent since the n b s and n s must continue to satisy the restrictions given in Eqs (-4. Since N and E are ixed, the variations in n b and n must satisy the ollowing equations: dn + dn = 0 (8 b = = and b = = ε dn + ε dn = 0 (9 Hence, along with Eq. (7, Eqs [(8 and (9] must also be satisied. These equations can be combined by the method o Lagrange s undetermined multipliers which are denoted by α and β. Thus, multiplying the irst and the second terms in Eq. (8 by ( α b and ( α, respectively, and Eq. (9 by ( β and adding to Eq. (7, we get, = ln( Cb ( αb + βε dnb nb + ln( Cb ( α + βε dn = 0 = n (0 Now Eq. (0 demands that all the terms should be separately equal to zero, and the terms or which dn b and dn are not equal to zero, then, the coeicients o dn b and dn should be, respectively, equal to zero. I we assume that one o the dn b s and dn s is non-zero, then the corresponding coeicients will be zero. Thus we can write, n n b ln( C ( α + βε =0 ( b b ln( C ( α + βε =0 ( b Eqs [( and (] are true or all values o. Substituting or C b rom Eq. (0 in Eq. ( gives, ω ( n nb = exp( αb + βε (3 n b Similarly, substituting or C b rom Eq. (0 in Eq. ( gives, ( ω n = exp( α + βε ( n n b (4 Eqs [(3 and (4] are solved or n b and n to get: nb = ω exp( αb α βε ω exp( αb α βε + exp( αb βε + ω exp( αb α βε (5 ω exp( αb α βε n = + exp( α b βε + ω exp( α b α βε (6 μ μ b where, α b =, α = and β = kt kt kt (7 Equation (5 gives the most probable distribution in energy or bosons in the ensemble, and Eq. (6 gives the most probable distribution in energy or ermions in the ensemble. In Eq. (7 μ b is the chemical potential or bosons and μ is the chemical potential or ermions. We should note that n is contained in the expression or n b. 4 Partition unction or the ensemble The general expression or the partition unction Q or an ensemble o bosons and ermions can be written as:

5 AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 753 μ μb Q= Nexp kt μ μb = ( nb + n exp (8 kt = 5 Results and Discussion The general expressions or entropy S, the internal energy E and the speciic heat C v are given as 3, lnq E= NkT T (30 Substituting or n b rom Eq. (5 and n rom Eq. (6 in Eq. (8, we get Q S = kn ln + T ln Q N T (3 ε Q= ω exp = kt μb+ μ ε + ω exp kt + μb ε μb+ μ ε + exp + ω exp kt kt (9 Equation (9 is the expression or the partition unction or a mixture o bosons and ermions in the ensemble, in which pair interaction between the two types o particles is considered. It must be understood that the partition unction Q has two signiicant terms: One term is μb+ μ ε exp kt This quantity contains ε which is a consequence o the pair interaction between the bosons and ermions. It can also be interpreted as the energy o a boson ermion pair in the -th energy level. The second term is, ln( Q ln( Q = NkT + NkT T V T C v V (3 Together with these equations, let us introduce the ermion concentration η given by η=n /N (33 or N=N /η (34 To perorm the calculations, we substituted or N and Q in the above thermodynamic relations rom Eqs [(5, (6 and (9]. The molar thermodynamic quantities are o interest. The experiments done by Chan et al. 4,7 mainly ocused on the molar quantities o liquid helium-3 and liquid helium-4. Wilks and Bett 8 give the molar volume o liquid helium-3 as 40.0 cm 3, and its molar density as 0.07 g cm 3 ; and the molar volume o liquid helium-4 as 8.0 cm 3 and its molar density as 0.4 g cm 3. This, thereore, means that the molar mass o liquid helium-3 is.80 g and that o liquid helium-4 is 3.9 g. Although the chemical potential should have temperature dependence, at low temperatures it assumes a nearly constant value given by the expression 8, μb ε exp kt π h 3N μ = m πv 3 (35 The absence o μ, chemical potential or ermions, in this quantity, is an indication that the distribution o bosons is not aected by how ermions are distributed. The number o bosons, the number o bosonermion pairs that shall be ormed, and the number o ermions shall determine how many ree or unattached ermions shall remain to be distributed among the available energy states. where, m is the molar mass, V is the molar volume and N is the number o particles in one mole, and this is Avogadro s number = particles mol. Substitution o the empirical data into Eq. (35 gives the chemical potential μ, or ermions as, 7 μ ev = (36 and or bosons as,

6 754 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER μ b ev = (37 Since the transition temperature 8 o liquid helium-3 is very much lower than the transition temperature o liquid helium-4, it is interesting to know what happens in the vicinity o the transition temperature o liquid helium-4. Furthermore, the temperature ranges used in experiments 4,5,7,8 on such mixtures are so much higher than the transition temperature o liquid helium-3, which is lower than.5 mk. However, given that both liquid helium-3 and liquid helium 4 are at the same temperature, they are considered to be in thermal equilibrium. 5. Calculation o partition unction Q Using the essential parameters or liquid helium-3 and liquid helium-4 given in Table, we calculated the partition unction Q, using Eq. (9 or the mixture in the temperature range.0 K to.30 K in steps o 0.0 K. Table gives the values o Q or the mixture at dierent temperatures. Figure depicts the variation o Q with temperature T. Figure shows that there is an exponential rise in the value o the partition unction Q. The reason or this kind o behaviour is that at very low temperatures there are very ew energy states 8 that can be occupied by the particles in the assembly. However, at higher temperatures, the number o energy levels available or particle occupation could be large, and these could be called the excited states o the assembly o particles. 5. Calculation o internal energy Equation (30 is used to calculate how the internal energy E varies with temperature using the parameters listed in Table. Table 3 gives the variation in the values o E with temperature in the range.0 K to.30 K in steps o 0.0K. To convert the values o E rom electronvolt (ev to Joules (J, E values are multiplied by a actor o The graph o internal energy variation against temperature is given in Fig.. Figure shows that there is a rise in the total internal energy with temperature. This is not unusual since or any given thermodynamic system, the higher the temperature, the higher should be the internal energy. The increase in internal energy tends to be exponential in the temperature range.0 K to.5 K but tends to assume a nearly constant value o Table Essential parameters or liquid helium-3 and helium-4 Parameter Liquid helium-3 Liquid helium-4 Volume ( cm Density (gcm Mass (g Chemical potential(ev Table Values o the partition unction against temperature T (K Q (T 0 particles approximately.50 kj as the temperature approaches.30 K. 5.3 Calculation o entropy S Equation (3 is used to calculate the variation o S with T. The values are given in Table 4 and the graph showing the variation o S with T is plotted in Fig. 3. The graph in Fig. 3 has the same shape as the one or the variation o E with T. The values o entropy increases with temperature ust like the internal energy. By deinition, entropy is a measure o the molecular disorder o any given system. Naturally, there should be greater molecular disorder at higher

7 AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 755 temperatures as opposed to lower temperatures. At higher temperatures molecules have higher vibrational energies, creating more disorder and hence more entropy. These trends are clearly depicted in Fig. 3. Entropy is also an extensive thermodynamic quantity. That means it is proportional to the number o particles in a system. In act it is meaningless to talk about the entropy o a single particle. Thereore, the ewer the particles the lower is the value o entropy and vice versa. The shape o the curve in Fig. 3 is in good agreement with the result obtained in Fig., where at lower temperatures there are ewer energy Fig. Variation o partition unction with temperature Table 3 Values o internal energy E at dierent temperatures T (K Internal Energy, E (J Fig. Variation o internal energy with temperature Table 4 Values o entropy S at dierent temperatures T (K Entropy, S (J/K

8 756 INDIAN J PURE & APPL PHYS, VOL. 4, OCTOBER 004 Table 5 Values o speciic heat ( 0 5 at dierent temperatures T (K Speciic heat, C (J/mol. K Fig. 3 Variation o entropy with temperature states or the distribution o particles. This means entropy is expected to be low at lower temperatures. 5.4 Calculation o speciic heat C v Equation (3 is used to calculate how the speciic heat at constant volume or a mixture o liquid 3 He and liquid 4 He varies with temperature T. Since the molar quantities o the constituents o the mixture are the ones that were o interest, the values o the speciic heat were multiplied by a actor o and divided by to convert the units o speciic heat rom ev kg K to J mol K. This is because rom Table, 6.7g is the mass o moles o the mixture and thus kg o the mixture is roughly equal to moles. Table 5 gives the values o C v at dierent temperatures. The shape o the speciic heat curve exhibits a lot o luctuations. Speciic heat values are too low in the temperature range rom.0 K to.09 K. One particular eature that was ound to be more interesting is that the highest peak o the curve occurred at.5 K. From here we see a signiicant change o phase at.5 K. This happens to be very near the λ-transition temperature 8,9 or liquid 4 He, which is.66 K. Below this temperature liquid 4 He becomes a superluid. However, experimental observations by Chan et al. 4,7,9 showed shits in the transition temperature at which peaks in the value o the speciic heat occurred. This can be accounted or due to the act that, experimentally, a highly porous material called aerogel was used to control the low o liquid 3 He into liquid 4 He and changes in the thermodynamic quantities o the mixture were observed or dierent liquid 3 He concentrations. However, our theoretical model assumes a bulk mixture, meaning without aerogel, o the two liquids. Furthermore, our calculations do not include the low properties o the two liquids, or instance, superluidity in liquid 4 He is supposed to disappear above a certain critical velocity. The normal-superluid phase transition in pure liquid 4 He is a second order phase transition, whereas the phase change in the mixture o liquid 3 He into liquid 4 He are characterised with a lot o luctuations with no discontinuity. This may be due to the act that in our case the atoms o the two liquids have not been considered to be entirely independent but exchange energy through pair interaction. 6 Conclusions Dierent authors -5 studied the statistical thermodynamics or a mixture o bosons and ermions by putting orth dierent models. In these models particles were considered independent or weakly

9 AYODO et al.: MICRO-CANONICAL ENSEMBLE FOR BOSONS & FERMIONS 757 Fig. 4 Variation o speciic heat ( 0 5 with temperature interacting. In our model, the bosons and ermions are supposed to be interacting via a pair interaction, and the whole assembly is supposed to be in thermal equilibrium. Furthermore, there are more ermions than bosons in this ensemble. Comparing the calculations presented in Tables -5; and the graphs presented in Figs (-4, with the corresponding results in Re. 5, in which bosons were more than ermions, the ollowing marked dierences in the shapes o the curves can be observed: (i Partition unction Q The partition unction in Re. 5 becomes roughly constant ater T. K, whereas, in the present calculation, Q varies exponentially ater T. K. This means that at higher temperatures the occupation o excited states increases and this is the basic character o ermion-dominated systems. (ii Internal Energy E The internal energy in Re. 5 becomes maximum around T.4 K and then decreases as the temperature increases. In the present calculations, the value o the internal energy smoothly increases as the temperature is increased, and then becomes constant ater T.3 K. This means that a ermion-dominated system behaves like an electron gas. (iii Speciic heat C v The shape o the speciic heat C v and that o the internal energy E is the same in Re. 5, whereas we ind that the speciic heat C v has maxima and minima, and the maximum value o C v is around.5 K.The shape o the C v curve is dierent rom the shape o the curve or the internal energy E. It should be acceptable that the speciic heat C v or a ermion dominated system will be dierent rom a boson dominated system. It is the Pauli exclusion principle that restricts the low o ermions rom one level to another as the temperature changes, whereas no such restriction exists or a boson dominated system. Thus, a ermion-dominated system may reuse to absorb heat resulting in a negative speciic heat. The actual transition temperature o the mixture is at.5 K, below which the whole mixture goes into the superluid state. The magnitudes o the thermodynamic quantities increases as the value o η increases and this is evident rom Eqs [(30 to (34]. The phase transition is one that is not smooth but is characterised by luctuations. Reerences Gentile G, Phys Rev Lett, 7 ( Medvedev M V, Phys Rev Lett, 78 ( Landau & Lishitz, Statistical physics, Vol.,Third edition (Pergamon Press, New York (98. 4 Chan M H, Blum K I & Murphy S Q, Phy Rev Lett, 6 ( Khanna K M & Ayodo Y K, Indian J Pure & Appl Phys, 4 ( Baierhein R, Thermal Physics, First edition (Cambridge university Press, Chan M H, Mulders N & Reppy J, Physics Today, 7 ( Wilks J & Bett D S, An Introduction to liquid helium, Second edition (Clarendon Press, Oxord \ ( Greenberg O W, Phys Rev Lett, 43 ( Khanna K M & Mehrotra S N, Physica, 8A ( Khanna K M, Statistical Mechanics and Many-Body Problems (Today and Tomorrow Publishers, New Delhi, (986.

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 112. Homework #4. Benjamin Stahl. February 2, 2015

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 112. Homework #4. Benjamin Stahl. February 2, 2015 UIERSIY OF CALIFORIA - SAA CRUZ DEPARME OF PHYSICS PHYS Homework #4 Benjamin Stahl February, 05 PROBLEM It is given that the heat absorbed by a mole o ideal gas in a uasi-static process in which both its

More information

8. INTRODUCTION TO STATISTICAL THERMODYNAMICS

8. INTRODUCTION TO STATISTICAL THERMODYNAMICS n * D n d Fluid z z z FIGURE 8-1. A SYSTEM IS IN EQUILIBRIUM EVEN IF THERE ARE VARIATIONS IN THE NUMBER OF MOLECULES IN A SMALL VOLUME, SO LONG AS THE PROPERTIES ARE UNIFORM ON A MACROSCOPIC SCALE 8. INTRODUCTION

More information

5. Internal energy: The total energy with a system.

5. Internal energy: The total energy with a system. CAPTER 6 TERMODYNAMICS Brie Summary o the chapter:. Thermodynamics: Science which deals with study o dierent orms o energy and quantitative relationship.. System & Surroundings: The part o universe or

More information

Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling

Accuracy of free-energy perturbation calculations in molecular simulation. I. Modeling JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 17 1 MAY 2001 ARTICLES Accuracy o ree-energy perturbation calculations in molecular simulation. I. Modeling Nandou Lu and David A. Koke a) Department o Chemical

More information

ENERGY ANALYSIS: CLOSED SYSTEM

ENERGY ANALYSIS: CLOSED SYSTEM ENERGY ANALYSIS: CLOSED SYSTEM A closed system can exchange energy with its surroundings through heat and work transer. In other words, work and heat are the orms that energy can be transerred across the

More information

Fractional exclusion statistics: A generalised Pauli principle

Fractional exclusion statistics: A generalised Pauli principle Fractional exclusion statistics: A generalised Pauli principle M.V.N. Murthy Institute of Mathematical Sciences, Chennai (murthy@imsc.res.in) work done with R. Shankar Indian Academy of Sciences, 27 Nov.

More information

5. Internal energy: The total energy with a system.

5. Internal energy: The total energy with a system. CBSE sample papers, Question papers, Notes or Class 6 to CAPTER 6 TERMODYNAMICS Brie Summary o the chapter:. Thermodynamics: Science which deals with study o dierent orms o energy and quantitative relationship..

More information

Solid Thermodynamics (1)

Solid Thermodynamics (1) Solid Thermodynamics (1) Class notes based on MIT OCW by KAN K.A.Nelson and MB M.Bawendi Statistical Mechanics 2 1. Mathematics 1.1. Permutation: - Distinguishable balls (numbers on the surface of the

More information

Curve Sketching. The process of curve sketching can be performed in the following steps:

Curve Sketching. The process of curve sketching can be performed in the following steps: Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points

More information

AP* Bonding & Molecular Structure Free Response Questions page 1

AP* Bonding & Molecular Structure Free Response Questions page 1 AP* Bonding & Molecular Structure Free Response Questions page 1 Essay Questions 1991 a) two points ΔS will be negative. The system becomes more ordered as two gases orm a solid. b) two points ΔH must

More information

Fluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs

Fluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr

More information

Quantum Grand Canonical Ensemble

Quantum Grand Canonical Ensemble Chapter 16 Quantum Grand Canonical Ensemble How do we proceed quantum mechanically? For fermions the wavefunction is antisymmetric. An N particle basis function can be constructed in terms of single-particle

More information

Circuit Complexity / Counting Problems

Circuit Complexity / Counting Problems Lecture 5 Circuit Complexity / Counting Problems April 2, 24 Lecturer: Paul eame Notes: William Pentney 5. Circuit Complexity and Uniorm Complexity We will conclude our look at the basic relationship between

More information

Atkins / Paula Physical Chemistry, 8th Edition. Chapter 16. Statistical thermodynamics 1: the concepts

Atkins / Paula Physical Chemistry, 8th Edition. Chapter 16. Statistical thermodynamics 1: the concepts Atkins / Paula Physical Chemistry, 8th Edition Chapter 16. Statistical thermodynamics 1: the concepts The distribution of molecular states 16.1 Configurations and weights 16.2 The molecular partition function

More information

6.730 Physics for Solid State Applications

6.730 Physics for Solid State Applications 6.730 Physics for Solid State Applications Lecture 25: Chemical Potential and Equilibrium Outline Microstates and Counting System and Reservoir Microstates Constants in Equilibrium Temperature & Chemical

More information

Fluctuations of Trapped Particles

Fluctuations of Trapped Particles Fluctuations of Trapped Particles M.V.N. Murthy with Muoi Tran and R.K. Bhaduri (McMaster) IMSc Chennai Department of Physics, University of Mysore, Nov 2005 p. 1 Ground State Fluctuations Ensembles in

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

There are eight problems on the exam. Do all of the problems. Show your work

There are eight problems on the exam. Do all of the problems. Show your work CHM 3400 Fundamentals o Physical Chemistry Final Exam April 23, 2012 There are eight problems on the exam. Do all o the problems. Show your work R = 0.08206 L. atm/mole. K N A = 6.022 x 10 23 R = 0.08314

More information

Effect Of Duo Fermion Spin On The Specific Heat And Entropy Of A Mixture Of Helium Isotopes

Effect Of Duo Fermion Spin On The Specific Heat And Entropy Of A Mixture Of Helium Isotopes Eect O Duo Fermion Spin On he Speciic Heat And Entropy O A Mixture O Helium Isotopes * S.M. Lusamama iaii University, P.O ox 1699-5000, ungoma, enya. simonlusamama@kiu.ac.ke *Author or orrespondence.w.

More information

Part I: Thin Converging Lens

Part I: Thin Converging Lens Laboratory 1 PHY431 Fall 011 Part I: Thin Converging Lens This eperiment is a classic eercise in geometric optics. The goal is to measure the radius o curvature and ocal length o a single converging lens

More information

CHAPTER 9 Statistical Physics

CHAPTER 9 Statistical Physics CHAPTER 9 Statistical Physics 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Historical Overview Maxwell Velocity Distribution Equipartition Theorem Maxwell Speed Distribution Classical and Quantum Statistics Fermi-Dirac

More information

Physics 2B Chapter 17 Notes - First Law of Thermo Spring 2018

Physics 2B Chapter 17 Notes - First Law of Thermo Spring 2018 Internal Energy o a Gas Work Done by a Gas Special Processes The First Law o Thermodynamics p Diagrams The First Law o Thermodynamics is all about the energy o a gas: how much energy does the gas possess,

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

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

Recitation: 10 11/06/03

Recitation: 10 11/06/03 Recitation: 10 11/06/03 Ensembles and Relation to T.D. It is possible to expand both sides of the equation with F = kt lnq Q = e βe i If we expand both sides of this equation, we apparently obtain: i F

More information

ε1 ε2 ε3 ε4 ε

ε1 ε2 ε3 ε4 ε Que : 1 a.) The total number of macro states are listed below, ε1 ε2 ε3 ε4 ε5 But all the macro states will not be possible because of given degeneracy levels. Therefore only possible macro states will

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

Roberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points

Roberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques

More information

Definition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.

Definition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series. 2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the

More information

Chapter 14. Ideal Bose gas Equation of state

Chapter 14. Ideal Bose gas Equation of state Chapter 14 Ideal Bose gas In this chapter, we shall study the thermodynamic properties of a gas of non-interacting bosons. We will show that the symmetrization of the wavefunction due to the indistinguishability

More information

2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction

2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction . ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties

More information

Elements of Statistical Mechanics

Elements of Statistical Mechanics Elements of Statistical Mechanics Thermodynamics describes the properties of macroscopic bodies. Statistical mechanics allows us to obtain the laws of thermodynamics from the laws of mechanics, classical

More information

PHYS 328 HOMEWORK 10-- SOLUTIONS

PHYS 328 HOMEWORK 10-- SOLUTIONS PHYS 328 HOMEWORK 10-- SOLUTIONS 1. We start by considering the ratio of the probability of finding the system in the ionized state to the probability of finding the system in the state of a neutral H

More information

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Content-Thermodynamics & Statistical Mechanics 1. Kinetic theory of gases..(1-13) 1.1 Basic assumption of kinetic theory 1.1.1 Pressure exerted by a gas 1.2 Gas Law for Ideal gases: 1.2.1 Boyle s Law 1.2.2

More information

L11.P1 Lecture 11. Quantum statistical mechanics: summary

L11.P1 Lecture 11. Quantum statistical mechanics: summary Lecture 11 Page 1 L11.P1 Lecture 11 Quantum statistical mechanics: summary At absolute zero temperature, a physical system occupies the lowest possible energy configuration. When the temperature increases,

More information

Identical Particles. Bosons and Fermions

Identical Particles. Bosons and Fermions Identical Particles Bosons and Fermions In Quantum Mechanics there is no difference between particles and fields. The objects which we refer to as fields in classical physics (electromagnetic field, field

More information

Definite Integral and the Gibbs Paradox

Definite Integral and the Gibbs Paradox Acta Polytechnica Hungarica ol. 8, No. 4, 0 Definite Integral and the Gibbs Paradox TianZhi Shi College of Physics, Electronics and Electrical Engineering, HuaiYin Normal University, HuaiAn, JiangSu, China,

More information

UNIVERSITY OF LONDON. BSc and MSci EXAMINATION 2005 DO NOT TURN OVER UNTIL TOLD TO BEGIN

UNIVERSITY OF LONDON. BSc and MSci EXAMINATION 2005 DO NOT TURN OVER UNTIL TOLD TO BEGIN UNIVERSITY OF LONDON BSc and MSci EXAMINATION 005 For Internal Students of Royal Holloway DO NOT UNTIL TOLD TO BEGIN PH610B: CLASSICAL AND STATISTICAL THERMODYNAMICS PH610B: CLASSICAL AND STATISTICAL THERMODYNAMICS

More information

Part II: Statistical Physics

Part II: Statistical Physics Chapter 7: Quantum Statistics SDSMT, Physics 2013 Fall 1 Introduction 2 The Gibbs Factor Gibbs Factor Several examples 3 Quantum Statistics From high T to low T From Particle States to Occupation Numbers

More information

Chemistry and the material world Unit 4, Lecture 4 Matthias Lein

Chemistry and the material world Unit 4, Lecture 4 Matthias Lein Chemistry and the material world 123.102 Unit 4, Lecture 4 Matthias Lein Gibbs ree energy Gibbs ree energy to predict the direction o a chemical process. Exergonic and endergonic reactions. Temperature

More information

Monatomic ideal gas: partition functions and equation of state.

Monatomic ideal gas: partition functions and equation of state. Monatomic ideal gas: partition functions and equation of state. Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Statistical Thermodynamics, MC260P105, Lecture 3,

More information

Solid State Device Fundamentals

Solid State Device Fundamentals Solid State Device Fundamentals ES 345 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 Oice 4101b 1 The ree electron model o metals The ree electron model o metals

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able FUNCTIONS Quadratic Functions......8 Absolute Value Functions.....48 Translations o Functions..57 Radical Functions...61 Eponential Functions...7 Logarithmic Functions......8 Cubic Functions......91 Piece-Wise

More information

PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical

PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Subject Paper No and Title Module No and Title Module Tag 6: PHYSICAL CHEMISTRY-II (Statistical 1: Introduction to Statistical CHE_P6_M1 TABLE OF CONTENTS 1. Learning Outcomes 2. Statistical Mechanics

More information

Physics 127a: Class Notes

Physics 127a: Class Notes Physics 127a: Class Notes Lecture 15: Statistical Mechanics of Superfluidity Elementary excitations/quasiparticles In general, it is hard to list the energy eigenstates, needed to calculate the statistical

More information

Thermal and Statistical Physics Department Exam Last updated November 4, L π

Thermal and Statistical Physics Department Exam Last updated November 4, L π Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =

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

PHYS3113, 3d year Statistical Mechanics Tutorial problems. Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions

PHYS3113, 3d year Statistical Mechanics Tutorial problems. Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions 1 PHYS3113, 3d year Statistical Mechanics Tutorial problems Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions Problem 1 The macrostate probability in an ensemble of N spins 1/2 is

More information

Sound Attenuation at High Temperatures in Pt

Sound Attenuation at High Temperatures in Pt Vol. 109 006) ACTA PHYSICA POLONICA A No. Sound Attenuation at High Temperatures in Pt R.K. Singh and K.K. Pandey H.C.P.G. College, Varanasi-1001, U.P., India Received October 4, 005) Ultrasonic attenuation

More information

APPENDIX 1 ERROR ESTIMATION

APPENDIX 1 ERROR ESTIMATION 1 APPENDIX 1 ERROR ESTIMATION Measurements are always subject to some uncertainties no matter how modern and expensive equipment is used or how careully the measurements are perormed These uncertainties

More information

( ) ( )( k B ( ) ( ) ( ) ( ) ( ) ( k T B ) 2 = ε F. ( ) π 2. ( ) 1+ π 2. ( k T B ) 2 = 2 3 Nε 1+ π 2. ( ) = ε /( ε 0 ).

( ) ( )( k B ( ) ( ) ( ) ( ) ( ) ( k T B ) 2 = ε F. ( ) π 2. ( ) 1+ π 2. ( k T B ) 2 = 2 3 Nε 1+ π 2. ( ) = ε /( ε 0 ). PHYS47-Statistical Mechanics and hermal Physics all 7 Assignment #5 Due on November, 7 Problem ) Problem 4 Chapter 9 points) his problem consider a system with density of state D / ) A) o find the ermi

More information

Statistical. mechanics

Statistical. mechanics CHAPTER 15 Statistical Thermodynamics 1: The Concepts I. Introduction. A. Statistical mechanics is the bridge between microscopic and macroscopic world descriptions of nature. Statistical mechanics macroscopic

More information

Removing the mystery of entropy and thermodynamics. Part 3

Removing the mystery of entropy and thermodynamics. Part 3 Removing the mystery of entropy and thermodynamics. Part 3 arvey S. Leff a,b Physics Department Reed College, Portland, Oregon USA August 3, 20 Introduction In Part 3 of this five-part article, [, 2] simple

More information

There are six problems on the exam. Do all of the problems. Show your work

There are six problems on the exam. Do all of the problems. Show your work CHM 3400 Fundamentals o Physical Chemistry First Hour Exam There are six problems on the exam. Do all o the problems. Show your work R = 0.08206 L. atm/mole. K N A = 6.022 x 10 23 R = 0.08314 L. bar/mole.

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

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS (Adopted on 4 June 203) (Adopted on 4 June 203) ANNEX 8 (Adopted on 4 June 203) MSC 92/26/Add. Annex 8, page THE MARITIME SAFETY COMMITTEE, RECALLING Article 28(b) o the Convention on the International

More information

Thermal Physics. 1) Thermodynamics: Relates heat + work with empirical (observed, not derived) properties of materials (e.g. ideal gas: PV = nrt).

Thermal Physics. 1) Thermodynamics: Relates heat + work with empirical (observed, not derived) properties of materials (e.g. ideal gas: PV = nrt). Thermal Physics 1) Thermodynamics: Relates heat + work with empirical (observed, not derived) properties of materials (e.g. ideal gas: PV = nrt). 2) Statistical Mechanics: Uses models (can be more complicated)

More information

Derivation of the Boltzmann Distribution

Derivation of the Boltzmann Distribution CLASSICAL CONCEPT REVIEW 7 Derivation of the Boltzmann Distribution Consider an isolated system, whose total energy is therefore constant, consisting of an ensemble of identical particles 1 that can exchange

More information

RATIONAL FUNCTIONS. Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions

RATIONAL FUNCTIONS. Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions RATIONAL FUNCTIONS Finding Asymptotes..347 The Domain....350 Finding Intercepts.....35 Graphing Rational Functions... 35 345 Objectives The ollowing is a list o objectives or this section o the workbook.

More information

Scattering of Solitons of Modified KdV Equation with Self-consistent Sources

Scattering of Solitons of Modified KdV Equation with Self-consistent Sources Commun. Theor. Phys. Beijing, China 49 8 pp. 89 84 c Chinese Physical Society Vol. 49, No. 4, April 5, 8 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources ZHANG Da-Jun and WU Hua

More information

2. Thermodynamics of native point defects in GaAs

2. Thermodynamics of native point defects in GaAs 2. Thermodynamics o native point deects in The totality o point deects in a crystal comprise those existing in a perectly chemically pure crystal, so called intrinsic deects, and those associated with

More information

ME 328 Machine Design Vibration handout (vibrations is not covered in text)

ME 328 Machine Design Vibration handout (vibrations is not covered in text) ME 38 Machine Design Vibration handout (vibrations is not covered in text) The ollowing are two good textbooks or vibrations (any edition). There are numerous other texts o equal quality. M. L. James,

More information

arxiv:physics/ v1 [physics.atom-ph] 30 Apr 2001

arxiv:physics/ v1 [physics.atom-ph] 30 Apr 2001 Collective Modes in a Dilute Bose-Fermi Mixture arxiv:physics/1489v1 [physics.atom-ph] 3 Apr 21 S. K. Yip Institute o Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan Abstract We here study the

More information

Physics 5153 Classical Mechanics. Solution by Quadrature-1

Physics 5153 Classical Mechanics. Solution by Quadrature-1 October 14, 003 11:47:49 1 Introduction Physics 5153 Classical Mechanics Solution by Quadrature In the previous lectures, we have reduced the number o eective degrees o reedom that are needed to solve

More information

Thermodynamics Heat & Work The First Law of Thermodynamics

Thermodynamics Heat & Work The First Law of Thermodynamics Thermodynamics Heat & Work The First Law o Thermodynamics Lana Sheridan De Anza College April 26, 2018 Last time more about phase changes work, heat, and the irst law o thermodynamics Overview P-V diagrams

More information

Introduction. Chapter The Purpose of Statistical Mechanics

Introduction. Chapter The Purpose of Statistical Mechanics Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for

More information

N independent electrons in a volume V (assuming periodic boundary conditions) I] The system 3 V = ( ) k ( ) i k k k 1/2

N independent electrons in a volume V (assuming periodic boundary conditions) I] The system 3 V = ( ) k ( ) i k k k 1/2 Lecture #6. Understanding the properties of metals: the free electron model and the role of Pauli s exclusion principle.. Counting the states in the E model.. ermi energy, and momentum. 4. DOS 5. ermi-dirac

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

The non-interacting Bose gas

The non-interacting Bose gas Chapter The non-interacting Bose gas Learning goals What is a Bose-Einstein condensate and why does it form? What determines the critical temperature and the condensate fraction? What changes for trapped

More information

Imperial College London BSc/MSci EXAMINATION May 2008 THERMODYNAMICS & STATISTICAL PHYSICS

Imperial College London BSc/MSci EXAMINATION May 2008 THERMODYNAMICS & STATISTICAL PHYSICS Imperial College London BSc/MSci EXAMINATION May 2008 This paper is also taken for the relevant Examination for the Associateship THERMODYNAMICS & STATISTICAL PHYSICS For Second-Year Physics Students Wednesday,

More information

Advanced Thermodynamics. Jussi Eloranta (Updated: January 22, 2018)

Advanced Thermodynamics. Jussi Eloranta (Updated: January 22, 2018) Advanced Thermodynamics Jussi Eloranta (jmeloranta@gmail.com) (Updated: January 22, 2018) Chapter 1: The machinery of statistical thermodynamics A statistical model that can be derived exactly from the

More information

( x) f = where P and Q are polynomials.

( x) f = where P and Q are polynomials. 9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational

More information

X-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity

X-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity X-ray Diraction Interaction o Waves Reciprocal Lattice and Diraction X-ray Scattering by Atoms The Integrated Intensity Basic Principles o Interaction o Waves Periodic waves characteristic: Frequency :

More information

21 Lecture 21: Ideal quantum gases II

21 Lecture 21: Ideal quantum gases II 2. LECTURE 2: IDEAL QUANTUM GASES II 25 2 Lecture 2: Ideal quantum gases II Summary Elementary low temperature behaviors of non-interacting particle systems are discussed. We will guess low temperature

More information

Two Constants of Motion in the Generalized Damped Oscillator

Two Constants of Motion in the Generalized Damped Oscillator Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 2, 57-65 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2016.511107 Two Constants o Motion in the Generalized Damped Oscillator

More information

THERMODYNAMICS OF A GRAND-CANONICAL BINARY SYSTEM AT LOW TEMPERATURES

THERMODYNAMICS OF A GRAND-CANONICAL BINARY SYSTEM AT LOW TEMPERATURES THERMODYNAMICS OF A GRAND-CANONICAL BINARY SYSTEM AT LOW TEMPERATURES *SakwaT.W 1, AyodoY.K, Sarai A. 1, Khanna K.M, Rapando B.W 4 & Mukoya A.K 1 1 Department of Physics, Masinde Muliro University of Science

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION DOI: 1.138/NPHYS2549 Electrically tunable transverse magnetic ocusing in graphene Supplementary Inormation Thiti Taychatanapat 1,2, Kenji Watanabe 3, Takashi Taniguchi 3, Pablo Jarillo-Herrero 2 1 Department

More information

Numerical Methods - Lecture 2. Numerical Methods. Lecture 2. Analysis of errors in numerical methods

Numerical Methods - Lecture 2. Numerical Methods. Lecture 2. Analysis of errors in numerical methods Numerical Methods - Lecture 1 Numerical Methods Lecture. Analysis o errors in numerical methods Numerical Methods - Lecture Why represent numbers in loating point ormat? Eample 1. How a number 56.78 can

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2014

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2014 Lecture 10 1/31/14 University o Washington Department o Chemistry Chemistry 453 Winter Quarter 014 A on-cooperative & Fully Cooperative inding: Scatchard & Hill Plots Assume binding siteswe have derived

More information

Anderson impurity in a semiconductor

Anderson impurity in a semiconductor PHYSICAL REVIEW B VOLUME 54, NUMBER 12 Anderson impurity in a semiconductor 15 SEPTEMBER 1996-II Clare C. Yu and M. Guerrero * Department o Physics and Astronomy, University o Caliornia, Irvine, Caliornia

More information

3. Photons and phonons

3. Photons and phonons Statistical and Low Temperature Physics (PHYS393) 3. Photons and phonons Kai Hock 2010-2011 University of Liverpool Contents 3.1 Phonons 3.2 Photons 3.3 Exercises Photons and phonons 1 3.1 Phonons Photons

More information

Lecture 2: Intro. Statistical Mechanics

Lecture 2: Intro. Statistical Mechanics Lecture 2: Intro. Statistical Mechanics Statistical mechanics: concepts Aims: A microscopic view of entropy: Joule expansion reviewed. Boltzmann s postulate. S k ln g. Methods: Calculating arrangements;

More information

Chem 406 Biophysical Chemistry Lecture 1 Transport Processes, Sedimentation & Diffusion

Chem 406 Biophysical Chemistry Lecture 1 Transport Processes, Sedimentation & Diffusion Chem 406 Biophysical Chemistry Lecture 1 Transport Processes, Sedimentation & Diusion I. Introduction A. There are a group o biophysical techniques that are based on transport processes. 1. Transport processes

More information

Lectures 16: Phase Transitions

Lectures 16: Phase Transitions Lectures 16: Phase Transitions Continuous Phase transitions Aims: Mean-field theory: Order parameter. Order-disorder transitions. Examples: β-brass (CuZn), Ferromagnetic transition in zero field. Universality.

More information

Statistical thermodynamics L1-L3. Lectures 11, 12, 13 of CY101

Statistical thermodynamics L1-L3. Lectures 11, 12, 13 of CY101 Statistical thermodynamics L1-L3 Lectures 11, 12, 13 of CY101 Need for statistical thermodynamics Microscopic and macroscopic world Distribution of energy - population Principle of equal a priori probabilities

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

Thermodynamics & Statistical Mechanics SCQF Level 9, U03272, PHY-3-ThermStat. Thursday 24th April, a.m p.m.

Thermodynamics & Statistical Mechanics SCQF Level 9, U03272, PHY-3-ThermStat. Thursday 24th April, a.m p.m. College of Science and Engineering School of Physics H T O F E E U D N I I N V E B R U S I R T Y H G Thermodynamics & Statistical Mechanics SCQF Level 9, U03272, PHY-3-ThermStat Thursday 24th April, 2008

More information

arxiv: v2 [cond-mat.supr-con] 22 Aug 2008

arxiv: v2 [cond-mat.supr-con] 22 Aug 2008 On Translational Superluidity and the Landau Criterion or Bose Gases in the Gross-Pitaevski Limit Walter F. Wreszinski Departamento de Física Matemática,Universidade de São Paulo, C.P. 66318-05315-970

More information

STATISTICAL MECHANICS

STATISTICAL MECHANICS STATISTICAL MECHANICS PD Dr. Christian Holm PART 0 Introduction to statistical mechanics -Statistical mechanics: is the tool to link macroscopic physics with microscopic physics (quantum physics). -The

More information

Differentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve.

Differentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve. Dierentiation The main problem o dierential calculus deals with inding the slope o the tangent line at a point on a curve. deinition() : The slope o a curve at a point p is the slope, i it eists, o the

More information

Lecture 2 The First Law of Thermodynamics (Ch.1)

Lecture 2 The First Law of Thermodynamics (Ch.1) Lecture he First Law o hermodynamics (h.) Lecture - we introduced macroscopic parameters that describe the state o a thermodynamic system (including temperature), the equation o state (,,) 0, and linked

More information

Physics 4311 ANSWERS: Sample Problems for Exam #2. (1)Short answer questions:

Physics 4311 ANSWERS: Sample Problems for Exam #2. (1)Short answer questions: (1)Short answer questions: Physics 4311 ANSWERS: Sample Problems for Exam #2 (a) Consider an isolated system that consists of several subsystems interacting thermally and mechanically with each other.

More information

Lecture 10: Condensed matter systems

Lecture 10: Condensed matter systems Lecture 10: Condensed matter systems Ideal quantum, condensed system of fermions Aims: Degenerate systems: non-interacting quantum particles at high density ermions: ermi energy and chemical potential

More information

5. Systems in contact with a thermal bath

5. Systems in contact with a thermal bath 5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)

More information

Section 4 Statistical Thermodynamics of delocalised particles

Section 4 Statistical Thermodynamics of delocalised particles Section 4 Statistical Thermodynamics of delocalised particles 4.1 Classical Ideal Gas 4.1.1 Indistinguishability Since the partition function is proportional to probabilities it follows that for composite

More information

Appendix 1: Normal Modes, Phase Space and Statistical Physics

Appendix 1: Normal Modes, Phase Space and Statistical Physics Appendix : Normal Modes, Phase Space and Statistical Physics The last line of the introduction to the first edition states that it is the wide validity of relatively few principles which this book seeks

More information

COMPARISON OF THERMAL CHARACTERISTICS BETWEEN THE PLATE-FIN AND PIN-FIN HEAT SINKS IN NATURAL CONVECTION

COMPARISON OF THERMAL CHARACTERISTICS BETWEEN THE PLATE-FIN AND PIN-FIN HEAT SINKS IN NATURAL CONVECTION HEFAT014 10 th International Conerence on Heat Transer, Fluid Mechanics and Thermodynamics 14 6 July 014 Orlando, Florida COMPARISON OF THERMA CHARACTERISTICS BETWEEN THE PATE-FIN AND PIN-FIN HEAT SINKS

More information

Lecture 25: Heat and The 1st Law of Thermodynamics Prof. WAN, Xin

Lecture 25: Heat and The 1st Law of Thermodynamics Prof. WAN, Xin General Physics I Lecture 5: Heat and he 1st Law o hermodynamics Pro. WAN, Xin xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Latent Heat in Phase Changes Latent Heat he latent heat o vaporization or

More information

Chapter 4: Properties of Pure Substances. Pure Substance. Phases of a Pure Substance. Phase-Change Processes of Pure Substances

Chapter 4: Properties of Pure Substances. Pure Substance. Phases of a Pure Substance. Phase-Change Processes of Pure Substances Chapter 4: roperties o ure Substances ure Substance A substance that has a ixed chemical composition throughout is called a pure substance such as water, air, and nitrogen A pure substance does not hae

More information

Math 1314 Lesson 23 Partial Derivatives

Math 1314 Lesson 23 Partial Derivatives Math 1314 Lesson 3 Partial Derivatives When we are asked to ind the derivative o a unction o a single variable, (x), we know exactly what to do However, when we have a unction o two variables, there is

More information