Page 1 Lecture 2: Review and preview Let us begin with a review of undergraduate statistical physics, looking back at what you have already learnt in your rst course on statistical physics. Both Newtonian mechanics and its quantum counterpart, the Heisenberg- Schrodinger wave-mechanics are inherently deterministic theories. Consider for instance, the Hamiltonian formulation of Newton's laws dp dt = @H(p;q) @q dq dt = @H(p;q) @p (1) In these equations, knowing the initial condition (p(t = 0);q(t = 0)) xes behaviour for all t > 0. The situation is similar in quantum mechanics. Consider Dirac's formulation of Schrodinger's equation i dj (t)i dt = Hj i (2) Knowing initial statej (t = 0)i xes behaviour for allt> 0 given the system HamiltonianH. Now, given the success of Newtonian mechanics (or its relativistic gener- alizations) in treating the behaviour of classical few-body systems and the corresponding success of the Schrodinger equation in the quantum realm, it might perhaps be natural to implicitly assume that the behaviour of large macroscopic bodies|their macroscopic properties, internal structure and dynamics|could all be understood at least in principle by an applica- tion of these deterministic laws to all the 10 23 atoms that make up the macroscopic body. However, the great insight which forms the foundation of Statistical Physics is that such an approach is neither feasible nor relevant! The rst point is that it is simply not feasible to follow trajectories of 10 23 electrons in a crystal or 10 23 atoms in a gas. The second, more fundamental point is that this in- formation on the trajectories of all the particles, or the time evolution of the many-body wavefunction, does not help us understand the microscopic sig- nicance of macroscopic notions like \temperature", \hotness" vs \coldness" etc, which are key ingredients in our description of macroscopic systems. These key ingredients are assembled to form the science of thermodynam- ics: As we know, thermodynamics starts with operational denitions for a 1 Page 2 Lecture 2: Review and preview Let us begin with a review of undergraduate statistical physics, looking back at what you have already learnt in your rst course on statistical physics. Both Newtonian mechanics and its quantum counterpart, the Heisenberg- Schrodinger wave-mechanics are inherently deterministic theories. Consider for instance, the Hamiltonian formulation of Newton's laws dp dt = @H(p;q) @q dq dt = @H(p;q) @p (1) In these equations, knowing the initial condition (p(t = 0);q(t = 0)) xes behaviour for all t > 0. The situation is similar in quantum mechanics. Consider Dirac's formulation of Schrodinger's equation i dj (t)i dt = Hj i (2) Knowing initial statej (t = 0)i xes behaviour for allt> 0 given the system HamiltonianH. Now, given the success of Newtonian mechanics (or its relativistic gener- alizations) in treating the behaviour of classical few-body systems and the corresponding success of the Schrodinger equation in the quantum realm, it might perhaps be natural to implicitly assume that the behaviour of large macroscopic bodies|their macroscopic properties, internal structure and dynamics|could all be understood at least in principle by an applica- tion of these deterministic laws to all the 10 23 atoms that make up the macroscopic body. However, the great insight which forms the foundation of Statistical Physics is that such an approach is neither feasible nor relevant! The rst point is that it is simply not feasible to follow trajectories of 10 23 electrons in a crystal or 10 23 atoms in a gas. The second, more fundamental point is that this in- formation on the trajectories of all the particles, or the time evolution of the many-body wavefunction, does not help us understand the microscopic sig- nicance of macroscopic notions like \temperature", \hotness" vs \coldness" etc, which are key ingredients in our description of macroscopic systems. These key ingredients are assembled to form the science of thermodynam- ics: As we know, thermodynamics starts with operational denitions for a 1 few key quantities, the so-called \thermodynamic variables" that characterize the state of a macroscopic body. These include the degree of \disorder" and quantity of heat, quantied by the thermodynamic entropyS, the hotness or coldness of macroscopic bodies, characterized by the absolute temperature T , the quantity of \available" energy, characterized by the free energyF , the internal energy U, and so on. The predictive power of thermodynamics derives from a few simple prop- erties these thermodynamic variables are postulated to satisfy. For instance, heat always ows from body A to B when in contact if T A > T B . S either increases or remains constant with time. And S goes to zero as T! 0, and so on. Now, as remarked earlier, it is by no means obvious at all where these properties like entropy and absolute temperature are \hiding" in the the trajectories of 10 23 atoms in a gas, or the evolution of the many-body wavefunction in a hilbert space of dimension 10 23 . And the central insight of statistical physics is the realization that these thermodynamic properties are emergent and statistical in nature. The emergent aspect has to do with the fact that it makes no sense to say a single atom is \hot" or \cold", or ascribe temperatureT to it. But one mole of the corresponding gas in equilibrium can be described by thermodynamics, and does have a well-dened temperature at least in equilibrium. Likewise, there is no precise sense in which there is a sensibly dened entropy for a system of few atoms. Entropy, and the Third Law of thermodynamics both re ects the properties and behaviour of macroscopically large collections of atoms. Similarly, there is no sense in which a few atoms of Helium are in a super uid state. Super uidity (and we will have much more to say about it later) is a property of a macroscopically large collection of Helium atoms. The statistical aspect is another facet which again re ects the key role played by the \thermodynamic limit", i.e. the limit of macroscopically large system sizes (formally dened by keeping the density xed and nite, but sending the volume to innity). It has to do with the fact that thermody- namic laws can be violated by rare uctuations in small systems. Thus, if you insist on using the operational denitions of thermodynamics to measure the entropy of very tiny systems, say a few dozen molecules bound together to form a polymer chain, you will nd that the third law of thermodynamics can be disobeyed by rare uctuations in the behaviour of the system. Consequently, our microscopic understanding of thermodynamic proper- ties has a statistical avour that you are already familiar with. For the sake 2 Page 3 Lecture 2: Review and preview Let us begin with a review of undergraduate statistical physics, looking back at what you have already learnt in your rst course on statistical physics. Both Newtonian mechanics and its quantum counterpart, the Heisenberg- Schrodinger wave-mechanics are inherently deterministic theories. Consider for instance, the Hamiltonian formulation of Newton's laws dp dt = @H(p;q) @q dq dt = @H(p;q) @p (1) In these equations, knowing the initial condition (p(t = 0);q(t = 0)) xes behaviour for all t > 0. The situation is similar in quantum mechanics. Consider Dirac's formulation of Schrodinger's equation i dj (t)i dt = Hj i (2) Knowing initial statej (t = 0)i xes behaviour for allt> 0 given the system HamiltonianH. Now, given the success of Newtonian mechanics (or its relativistic gener- alizations) in treating the behaviour of classical few-body systems and the corresponding success of the Schrodinger equation in the quantum realm, it might perhaps be natural to implicitly assume that the behaviour of large macroscopic bodies|their macroscopic properties, internal structure and dynamics|could all be understood at least in principle by an applica- tion of these deterministic laws to all the 10 23 atoms that make up the macroscopic body. However, the great insight which forms the foundation of Statistical Physics is that such an approach is neither feasible nor relevant! The rst point is that it is simply not feasible to follow trajectories of 10 23 electrons in a crystal or 10 23 atoms in a gas. The second, more fundamental point is that this in- formation on the trajectories of all the particles, or the time evolution of the many-body wavefunction, does not help us understand the microscopic sig- nicance of macroscopic notions like \temperature", \hotness" vs \coldness" etc, which are key ingredients in our description of macroscopic systems. These key ingredients are assembled to form the science of thermodynam- ics: As we know, thermodynamics starts with operational denitions for a 1 few key quantities, the so-called \thermodynamic variables" that characterize the state of a macroscopic body. These include the degree of \disorder" and quantity of heat, quantied by the thermodynamic entropyS, the hotness or coldness of macroscopic bodies, characterized by the absolute temperature T , the quantity of \available" energy, characterized by the free energyF , the internal energy U, and so on. The predictive power of thermodynamics derives from a few simple prop- erties these thermodynamic variables are postulated to satisfy. For instance, heat always ows from body A to B when in contact if T A > T B . S either increases or remains constant with time. And S goes to zero as T! 0, and so on. Now, as remarked earlier, it is by no means obvious at all where these properties like entropy and absolute temperature are \hiding" in the the trajectories of 10 23 atoms in a gas, or the evolution of the many-body wavefunction in a hilbert space of dimension 10 23 . And the central insight of statistical physics is the realization that these thermodynamic properties are emergent and statistical in nature. The emergent aspect has to do with the fact that it makes no sense to say a single atom is \hot" or \cold", or ascribe temperatureT to it. But one mole of the corresponding gas in equilibrium can be described by thermodynamics, and does have a well-dened temperature at least in equilibrium. Likewise, there is no precise sense in which there is a sensibly dened entropy for a system of few atoms. Entropy, and the Third Law of thermodynamics both re ects the properties and behaviour of macroscopically large collections of atoms. Similarly, there is no sense in which a few atoms of Helium are in a super uid state. Super uidity (and we will have much more to say about it later) is a property of a macroscopically large collection of Helium atoms. The statistical aspect is another facet which again re ects the key role played by the \thermodynamic limit", i.e. the limit of macroscopically large system sizes (formally dened by keeping the density xed and nite, but sending the volume to innity). It has to do with the fact that thermody- namic laws can be violated by rare uctuations in small systems. Thus, if you insist on using the operational denitions of thermodynamics to measure the entropy of very tiny systems, say a few dozen molecules bound together to form a polymer chain, you will nd that the third law of thermodynamics can be disobeyed by rare uctuations in the behaviour of the system. Consequently, our microscopic understanding of thermodynamic proper- ties has a statistical avour that you are already familiar with. For the sake 2 of completeness, we provide a quick review: The basic idea is to start with the Gibbs distribution function, which postulates that a a macroscopic system of N particles in xed volume V is in eigenstatejmi with probability P m = 1 Z exp(E m (V;N)=k B T ) where Z = X m exp(E m (V;N)=k B T ) (3) Here, Z is the canonical partition function of the system. An interesting aspect of this statistical description is that T , an emergent property of a macroscopic system, enters in the relative probabilities of various m. With this starting point, one denes U = X m E m P m (4) F UTS =k B T log(Z) (5) where F is the Helmholtz free energy and U the internal energy. The tem- perature T , the internal energy U, the entropy S (dened implicitly in the above by subtracting the rst equation from the second) and the free en- ergy F dened in this manner are then argued to be posssessed of all the properties one expects of the corresponding quantities dened operationally in thermodynamics. This provides an a posteriori justication of the Gibbs distribution function. This framework generalizes readily if one wants a more general prescrip- tion that allows for number uctuations. One starts with a larger space of states which considers all possible values of the total number N, and postu- lates the grand-canonical distribution function P m;N = 1 Z GC exp ((E m (V;N)N)=k B T ) Z GC = X m;N exp ((E m (V;N)N)=k B T ) (6) Z GC , the grand-canonical partition function depends on the chemical poten- tial , which can be thought of as the energy cost of adding a particle. Note that is an \intensive" variable, which can be thought of as a \Lagrange- multiplier" that xes the mean number of particles to equal what we expect 3 Page 4 Lecture 2: Review and preview Let us begin with a review of undergraduate statistical physics, looking back at what you have already learnt in your rst course on statistical physics. Both Newtonian mechanics and its quantum counterpart, the Heisenberg- Schrodinger wave-mechanics are inherently deterministic theories. Consider for instance, the Hamiltonian formulation of Newton's laws dp dt = @H(p;q) @q dq dt = @H(p;q) @p (1) In these equations, knowing the initial condition (p(t = 0);q(t = 0)) xes behaviour for all t > 0. The situation is similar in quantum mechanics. Consider Dirac's formulation of Schrodinger's equation i dj (t)i dt = Hj i (2) Knowing initial statej (t = 0)i xes behaviour for allt> 0 given the system HamiltonianH. Now, given the success of Newtonian mechanics (or its relativistic gener- alizations) in treating the behaviour of classical few-body systems and the corresponding success of the Schrodinger equation in the quantum realm, it might perhaps be natural to implicitly assume that the behaviour of large macroscopic bodies|their macroscopic properties, internal structure and dynamics|could all be understood at least in principle by an applica- tion of these deterministic laws to all the 10 23 atoms that make up the macroscopic body. However, the great insight which forms the foundation of Statistical Physics is that such an approach is neither feasible nor relevant! The rst point is that it is simply not feasible to follow trajectories of 10 23 electrons in a crystal or 10 23 atoms in a gas. The second, more fundamental point is that this in- formation on the trajectories of all the particles, or the time evolution of the many-body wavefunction, does not help us understand the microscopic sig- nicance of macroscopic notions like \temperature", \hotness" vs \coldness" etc, which are key ingredients in our description of macroscopic systems. These key ingredients are assembled to form the science of thermodynam- ics: As we know, thermodynamics starts with operational denitions for a 1 few key quantities, the so-called \thermodynamic variables" that characterize the state of a macroscopic body. These include the degree of \disorder" and quantity of heat, quantied by the thermodynamic entropyS, the hotness or coldness of macroscopic bodies, characterized by the absolute temperature T , the quantity of \available" energy, characterized by the free energyF , the internal energy U, and so on. The predictive power of thermodynamics derives from a few simple prop- erties these thermodynamic variables are postulated to satisfy. For instance, heat always ows from body A to B when in contact if T A > T B . S either increases or remains constant with time. And S goes to zero as T! 0, and so on. Now, as remarked earlier, it is by no means obvious at all where these properties like entropy and absolute temperature are \hiding" in the the trajectories of 10 23 atoms in a gas, or the evolution of the many-body wavefunction in a hilbert space of dimension 10 23 . And the central insight of statistical physics is the realization that these thermodynamic properties are emergent and statistical in nature. The emergent aspect has to do with the fact that it makes no sense to say a single atom is \hot" or \cold", or ascribe temperatureT to it. But one mole of the corresponding gas in equilibrium can be described by thermodynamics, and does have a well-dened temperature at least in equilibrium. Likewise, there is no precise sense in which there is a sensibly dened entropy for a system of few atoms. Entropy, and the Third Law of thermodynamics both re ects the properties and behaviour of macroscopically large collections of atoms. Similarly, there is no sense in which a few atoms of Helium are in a super uid state. Super uidity (and we will have much more to say about it later) is a property of a macroscopically large collection of Helium atoms. The statistical aspect is another facet which again re ects the key role played by the \thermodynamic limit", i.e. the limit of macroscopically large system sizes (formally dened by keeping the density xed and nite, but sending the volume to innity). It has to do with the fact that thermody- namic laws can be violated by rare uctuations in small systems. Thus, if you insist on using the operational denitions of thermodynamics to measure the entropy of very tiny systems, say a few dozen molecules bound together to form a polymer chain, you will nd that the third law of thermodynamics can be disobeyed by rare uctuations in the behaviour of the system. Consequently, our microscopic understanding of thermodynamic proper- ties has a statistical avour that you are already familiar with. For the sake 2 of completeness, we provide a quick review: The basic idea is to start with the Gibbs distribution function, which postulates that a a macroscopic system of N particles in xed volume V is in eigenstatejmi with probability P m = 1 Z exp(E m (V;N)=k B T ) where Z = X m exp(E m (V;N)=k B T ) (3) Here, Z is the canonical partition function of the system. An interesting aspect of this statistical description is that T , an emergent property of a macroscopic system, enters in the relative probabilities of various m. With this starting point, one denes U = X m E m P m (4) F UTS =k B T log(Z) (5) where F is the Helmholtz free energy and U the internal energy. The tem- perature T , the internal energy U, the entropy S (dened implicitly in the above by subtracting the rst equation from the second) and the free en- ergy F dened in this manner are then argued to be posssessed of all the properties one expects of the corresponding quantities dened operationally in thermodynamics. This provides an a posteriori justication of the Gibbs distribution function. This framework generalizes readily if one wants a more general prescrip- tion that allows for number uctuations. One starts with a larger space of states which considers all possible values of the total number N, and postu- lates the grand-canonical distribution function P m;N = 1 Z GC exp ((E m (V;N)N)=k B T ) Z GC = X m;N exp ((E m (V;N)N)=k B T ) (6) Z GC , the grand-canonical partition function depends on the chemical poten- tial , which can be thought of as the energy cost of adding a particle. Note that is an \intensive" variable, which can be thought of as a \Lagrange- multiplier" that xes the mean number of particles to equal what we expect 3 for a system of that average density. From the grand-canonical partition sum, one obtains another thermodynamic potential = k B T logZ GC (7) known as the Gibbs Free energy. If appropriate for the experimental situation at hand, one can also work with a distribution function that allows for a variable volume P m;V = 1 Z P exp ((E m (V;N) +PV )=k B T ) Z P = X m Z dV exp ((E m (V;N) +PV )=k B T ) (8) Here Z P is the partition function at xed pressure P |again, the intensive pressure variable can be thought of as a \Lagrange-multiplier" that xes the mean volume, and the corresponding thermodynamic potential H = k B T log(Z P ) (9) is the thermodynamic enthalpy. . Thus, the Gibbs distribution provides a statistical way of \understand- ing" the underlying rationale for the macroscopic laws of thermodynamics, and provides a clear calculational prescription for macroscopic concepts like temperature, free or available energy, entropy etc. Undergraduate treatments of statistical physics thus end on the following triumphant note: Macroscopic phenomena are governed by thermodynamics. Since statisti- cal mechanics provides the rationale for thermodynamics, all these phenom- ena can in principle be derived from statistical mechanics. At this point in our discussion, it is therefore worth asking: Really? Is this really true? More precisely, let us list some phenomenological facts, drawn from ev- eryday experience and a study of undergraduate physics, and ask: Where is all this lurking inside the Gibbs distribution function? Matter exists in several dierent phases: Crystalline solid, liquid, gaseous phases of H 2 O; ferromagnetic and paramagnetic metals, insulators... These phases are separated by phase transitions, accessed by changing pressure, temperature, magnetic eld... 4 Page 5 Lecture 2: Review and preview Let us begin with a review of undergraduate statistical physics, looking back at what you have already learnt in your rst course on statistical physics. Both Newtonian mechanics and its quantum counterpart, the Heisenberg- Schrodinger wave-mechanics are inherently deterministic theories. Consider for instance, the Hamiltonian formulation of Newton's laws dp dt = @H(p;q) @q dq dt = @H(p;q) @p (1) In these equations, knowing the initial condition (p(t = 0);q(t = 0)) xes behaviour for all t > 0. The situation is similar in quantum mechanics. Consider Dirac's formulation of Schrodinger's equation i dj (t)i dt = Hj i (2) Knowing initial statej (t = 0)i xes behaviour for allt> 0 given the system HamiltonianH. Now, given the success of Newtonian mechanics (or its relativistic gener- alizations) in treating the behaviour of classical few-body systems and the corresponding success of the Schrodinger equation in the quantum realm, it might perhaps be natural to implicitly assume that the behaviour of large macroscopic bodies|their macroscopic properties, internal structure and dynamics|could all be understood at least in principle by an applica- tion of these deterministic laws to all the 10 23 atoms that make up the macroscopic body. However, the great insight which forms the foundation of Statistical Physics is that such an approach is neither feasible nor relevant! The rst point is that it is simply not feasible to follow trajectories of 10 23 electrons in a crystal or 10 23 atoms in a gas. The second, more fundamental point is that this in- formation on the trajectories of all the particles, or the time evolution of the many-body wavefunction, does not help us understand the microscopic sig- nicance of macroscopic notions like \temperature", \hotness" vs \coldness" etc, which are key ingredients in our description of macroscopic systems. These key ingredients are assembled to form the science of thermodynam- ics: As we know, thermodynamics starts with operational denitions for a 1 few key quantities, the so-called \thermodynamic variables" that characterize the state of a macroscopic body. These include the degree of \disorder" and quantity of heat, quantied by the thermodynamic entropyS, the hotness or coldness of macroscopic bodies, characterized by the absolute temperature T , the quantity of \available" energy, characterized by the free energyF , the internal energy U, and so on. The predictive power of thermodynamics derives from a few simple prop- erties these thermodynamic variables are postulated to satisfy. For instance, heat always ows from body A to B when in contact if T A > T B . S either increases or remains constant with time. And S goes to zero as T! 0, and so on. Now, as remarked earlier, it is by no means obvious at all where these properties like entropy and absolute temperature are \hiding" in the the trajectories of 10 23 atoms in a gas, or the evolution of the many-body wavefunction in a hilbert space of dimension 10 23 . And the central insight of statistical physics is the realization that these thermodynamic properties are emergent and statistical in nature. The emergent aspect has to do with the fact that it makes no sense to say a single atom is \hot" or \cold", or ascribe temperatureT to it. But one mole of the corresponding gas in equilibrium can be described by thermodynamics, and does have a well-dened temperature at least in equilibrium. Likewise, there is no precise sense in which there is a sensibly dened entropy for a system of few atoms. Entropy, and the Third Law of thermodynamics both re ects the properties and behaviour of macroscopically large collections of atoms. Similarly, there is no sense in which a few atoms of Helium are in a super uid state. Super uidity (and we will have much more to say about it later) is a property of a macroscopically large collection of Helium atoms. The statistical aspect is another facet which again re ects the key role played by the \thermodynamic limit", i.e. the limit of macroscopically large system sizes (formally dened by keeping the density xed and nite, but sending the volume to innity). It has to do with the fact that thermody- namic laws can be violated by rare uctuations in small systems. Thus, if you insist on using the operational denitions of thermodynamics to measure the entropy of very tiny systems, say a few dozen molecules bound together to form a polymer chain, you will nd that the third law of thermodynamics can be disobeyed by rare uctuations in the behaviour of the system. Consequently, our microscopic understanding of thermodynamic proper- ties has a statistical avour that you are already familiar with. For the sake 2 of completeness, we provide a quick review: The basic idea is to start with the Gibbs distribution function, which postulates that a a macroscopic system of N particles in xed volume V is in eigenstatejmi with probability P m = 1 Z exp(E m (V;N)=k B T ) where Z = X m exp(E m (V;N)=k B T ) (3) Here, Z is the canonical partition function of the system. An interesting aspect of this statistical description is that T , an emergent property of a macroscopic system, enters in the relative probabilities of various m. With this starting point, one denes U = X m E m P m (4) F UTS =k B T log(Z) (5) where F is the Helmholtz free energy and U the internal energy. The tem- perature T , the internal energy U, the entropy S (dened implicitly in the above by subtracting the rst equation from the second) and the free en- ergy F dened in this manner are then argued to be posssessed of all the properties one expects of the corresponding quantities dened operationally in thermodynamics. This provides an a posteriori justication of the Gibbs distribution function. This framework generalizes readily if one wants a more general prescrip- tion that allows for number uctuations. One starts with a larger space of states which considers all possible values of the total number N, and postu- lates the grand-canonical distribution function P m;N = 1 Z GC exp ((E m (V;N)N)=k B T ) Z GC = X m;N exp ((E m (V;N)N)=k B T ) (6) Z GC , the grand-canonical partition function depends on the chemical poten- tial , which can be thought of as the energy cost of adding a particle. Note that is an \intensive" variable, which can be thought of as a \Lagrange- multiplier" that xes the mean number of particles to equal what we expect 3 for a system of that average density. From the grand-canonical partition sum, one obtains another thermodynamic potential = k B T logZ GC (7) known as the Gibbs Free energy. If appropriate for the experimental situation at hand, one can also work with a distribution function that allows for a variable volume P m;V = 1 Z P exp ((E m (V;N) +PV )=k B T ) Z P = X m Z dV exp ((E m (V;N) +PV )=k B T ) (8) Here Z P is the partition function at xed pressure P |again, the intensive pressure variable can be thought of as a \Lagrange-multiplier" that xes the mean volume, and the corresponding thermodynamic potential H = k B T log(Z P ) (9) is the thermodynamic enthalpy. . Thus, the Gibbs distribution provides a statistical way of \understand- ing" the underlying rationale for the macroscopic laws of thermodynamics, and provides a clear calculational prescription for macroscopic concepts like temperature, free or available energy, entropy etc. Undergraduate treatments of statistical physics thus end on the following triumphant note: Macroscopic phenomena are governed by thermodynamics. Since statisti- cal mechanics provides the rationale for thermodynamics, all these phenom- ena can in principle be derived from statistical mechanics. At this point in our discussion, it is therefore worth asking: Really? Is this really true? More precisely, let us list some phenomenological facts, drawn from ev- eryday experience and a study of undergraduate physics, and ask: Where is all this lurking inside the Gibbs distribution function? Matter exists in several dierent phases: Crystalline solid, liquid, gaseous phases of H 2 O; ferromagnetic and paramagnetic metals, insulators... These phases are separated by phase transitions, accessed by changing pressure, temperature, magnetic eld... 4 Some of these phases are \distinctly" dierent from other phases: Atoms in a crystal have a very \ordered" arrangement. Not so in a liquid Some phase transitions are \rst order" with latent heat of phase change. Other transitions are accompanied by large scale uctuations e.g. as evidenced by the phenomenon of critical opalescence at liquid- gas critical point. None of these follow in any automatic way from the basic prescription of Gibbs. They are all emergent properties of macroscopic aggregates of constituent particles, and require new ways of thinking to understand them well. One actually needs another layer of new concepts needed to \eciently" think about these macroscopic phenomena. Here, we list them by way of preview, and discuss them in some detail in the next lecture: Spontaneous breaking of symmetry. Phases distinguished by long-range order. Order parameters. Rigidity. Broken ergodicity. Gapless elementary excitations related to the underlying rigidity. More on this in the next lecture... 5Read More

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