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                <h1>Part I- Thermodynamics</h1>
                <h4>May 2019</h4>
            <hr>
            <div class="in-content">

                <h5>Index</h5>
                
                <ol type='i'>
                    <li><a href="#intro">Introduction</a></li>
                    <li><a href="#zero">The "Zeroeth" Law</a></li>
                    <li><a href="#energy">Heat, Work, Energy: The First Law</a></li>
                    <li><a href="ideal">The Ideal Gas Law</a></li>
                    <li><a href="states">Microstates and Macrostates</a></li>
                    <li><a href="#entropy">Entropy</a></li>
                </ol>

                <h5 id = "intro"> Section 1.1 - Introduction</h5>

                <p>
					This is going to be the first in a series of posts regarding
                    black hole thermodynamics, which, broadly speaking is my own
                    personal area of interest. In this series I will attempt to
                    give a broad (though technical) overview of the field and 
                    elaborate on many ofthe particulars as they come up. 
				</p>

                <p>
                    But before we dive in, it becomes us to elaborate an 
                    overview of the field. In brief, black hole thermodynamics
                    is all about the interplay between gravitational physics
                    and quantum field theory- two fields which even the 
                    most amateur physicist is likely to know do not tend to 
                    agree. Despite the two fields being supremely accurate 
                    descriptions of natural phenomena at certain scales, 
                    attempting to apply the laws of one to the other leads 
                    frequently to disastorous results.
                </p>

                <p>
                    In a region of space at or near the event horizon of a black
                    hole, the mingling ofthe two lines of thought are simply
                    unavoidable. For in the event of a strong vacuum fluctuation
                    in such a region, in such circumstances so that one of the
                    pairs produced in that phenomena is captured by the black
                    hole and the other is not, we are left with the recourse
                    that the black hole itself must "radiate" in some capacity-
                    though the particulars of such radiation are open to a
                    certain level of interpretation. However, the reader can
                    begin to see now the potential links between this field and
                    quantum gravity (and we will see links to quantum
                    information theory too- all in due time!)
                </p>

                <p>
                    Speaking, conveniently, of particulars, one of the first 
                    things that needs to be covered in a ground-up approach is 
                    elementary thermal physics. So in the spirit of walking 
                    before we run, lets take some time to cover the necessities 
                    in that field.
                </p>

                <h5 id="zero"> Section 1.2 - The Zeroeth Law </h5>

                <p>
                    The most important concept to discuss in this article is 
                    entropy- we will see later that there are strong parallels
                    between thermodynamic entropy and informational entropy that
                    motivate much of the work in the field. However,before we 
                    touch on this idea, we should cover even more elementary
                    material: the zeroeth and first laws of thermodynamics. 
                </p>

                <p>
                    Like all sane people, we're going to index at zero and start
                    with that law. The zeroeth law of thermodynamics is fairly
                    intuitive.
                </p>
                    
                <p>
                    <b>ZEROETH LAW:</b>If two thermal systems are
                    in thermal equilibrium with a third system, then they
                    are in thermal equilibrium with each other. 
                </p>

                <p>
                    For the mathematicians in the room, this is essentially a
                    statement asserting that thermal equilibrium obeys the 
                    transitive property. Further, we consider it obvious that 
                    the notion of thermal equilibrium is reflexive and 
                    symmetric- which makes thermal equilibrium an equivalence
                    relation. We call elements belonging to a given partition of
                    this equivalence class "isotherms".
                </p>
                    
                 <p>
                    That's all well and good, but we're still under the hood of
                    some jargon- what exactly is thermal equilibrium? In order
                    to answer that, I'm going to have to introduce some more
                    jargon, but if you can stick through it, you'll understand
                    soon enough.
                </p>

                <p>
                    We say systems are in thermal equilibrium if, on average, no
                    thermal energy is being transfered between them. We say on
                    average because in the real world, quantum mechanics plays
                    a part in physics and will in fact result in minor changes
                    to levels of thermal energy at an instant in time- but in
                    the long run these average to zero. We've traded one term
                    for another: we now need to ask what thermal energy is.
                </p>
                
                <p>
                    I'm so glad you asked.
                </p>

                <h5 id = "energy">Section 1.3 - Heat, Work, and Energy: The First Law</h5>

                <p>
                    Thermal energy is difficult to explain directly because
                    energy itself bears some explanation- the energy of a system
                    may be thought of as that body's capacity to do work: to
                    apply some force perpendicular to some distance traveled in
                    a given direction. Energy itself takes form in, broadly
                    speaking, two ways: kinetic, the energy due to the motion of
                    a body represented for our purposes by $T$ and given by 
                    some variation of the quadratic form $mv^2$ (we also see 
                    rotational and vibrational modes of kinetic energy- they too
                    have a quadratic form. More on them later.) We also have 
                    potential energies, which are dependent on a body's relation
                    to other bodies in a positional sense. We denote potential 
                    energy with $V$ and write that the total energy, $U$, is 
                    given by the equation $$U = T + V.$$ 
                    
                    In thermodynamics, we have two ways for changes in energy to
                    express themselves: in the forms of work, being energy 
                    transmitted actively and quantifiably via some external 
                    mechanism- the cannonical example being the compressive 
                    action of a piston on a cylindrical container. We often 
                    denote work with the symbol $W$. 
                </p>

                <p>
                    We also have a notion of spontaneous energy changes is
                    systems, which we call "heat", and represent with the 
                    symbol $Q$. This spontaneity is key- the energy transfer
                    not arise out of the external action of some mechanism, or
                    we would have to classify it as work. There are three
                    well-described ways that heat may be transferred: through
                    <a href =https://en.wikipedia.org/wiki/Thermal_conduction>conduction</a>,
                    <a href =https://en.wikipedia.org/wiki/Convective_heat_transfer>convection</a>,
                    and  <a href =https://en.wikipedia.org/wiki/Thermal_radiation>radiation</a>.
                    The particulars of each process are not material at the
                    moment, and I will leave it to the curious to familiarize
                    themselves with the notions if they are not conversant
                    already. They bare mentioning now because we will need some
                    understanding of radiative processes later. 
                </p>

                <p>
                    As we might say the total energy in a dynamical system is
                    $U = T + V$, we say that the total change in energy in a 
                    thermal system is give by $\Delta U = W + Q$. This is 
                    essentially a statement of the first law of thermodynamics,
                    but we might as well make that explicit.
                </p>
                
                <p>
                    <b>FIRST LAW:</b> The total energy of an isolated system 
                    (i.e. any system free from outside interaction) is conserved
                    and the changes in energy of a closed system (i.e. one free
                    from exchanges in matter but free to exchange energy) are 
                    given by $$\Delta U = W + Q$$ following the IUPAC conention.
                </p>

                <p>
                    At this point, it is necessary to introduce the idea of 
                    temperature and its relationship to thermal energy. To do
                    that, I want to introduce yet another element into the mix.
                </p>
                
                <h5 id="ideal">1.4 - The Ideal Gas Law

                <p>
                    The ideal gas law is a result that predicts certain 
                    properties of gasses in "ideal" conditions- in which the 
                    particles in the gas are moving fairly quickly (but not too
                    quickly) and are in general not close enough together to
                    be succeptible to interactions with the other particles. 
                    It was first concieved of as an amalgamation of many results
                    chemists had found in the early 19th century, but may also
                    be thought of as a result 
                    <a href="/articles/appendices/bht-a1/">derived</a> from the 
                    <a href="https://en.wikipedia.org/wiki/Equipartition_theorem">equipartition theorem</a>.
                    One's perspective relies on where they start and what they
                    take to be a first principle; I was taught this way and so
                    that's the way I'm going to present the material. 
                </p>
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                <h1>Part I- Thermodynamics</h1>
                <h4>May 2019</h4>
            <hr>
            <div class="in-content">

                <h5>Index</h5>
                
                <ol type='i'>
                    <li><a href="#intro">Introduction</a></li>
                    <li><a href="#zero">The "Zeroeth" Law</a></li>
                    <li><a href="#energy">Heat, Work, Energy: The First Law</a></li>
                    <li><a href="ideal">The Ideal Gas Law</a></li>
                    <li><a href="states">Microstates and Macrostates</a></li>
                    <li><a href="#entropy">Entropy</a></li>
                </ol>

                <h5 id = "intro"> Section 1.1 - Introduction</h5>

                <p>
					This is going to be the first in a series of posts regarding
                    black hole thermodynamics, which, broadly speaking is my own
                    personal area of interest. In this series I will attempt to
                    give a broad (though technical) overview of the field and 
                    elaborate on many ofthe particulars as they come up. 
				</p>

                <p>
                    But before we dive in, it becomes us to elaborate an 
                    overview of the field. In brief, black hole thermodynamics
                    is all about the interplay between gravitational physics
                    and quantum field theory- two fields which even the 
                    most amateur physicist is likely to know do not tend to 
                    agree. Despite the two fields being supremely accurate 
                    descriptions of natural phenomena at certain scales, 
                    attempting to apply the laws of one to the other leads 
                    frequently to disastorous results.
                </p>

                <p>
                    In a region of space at or near the event horizon of a black
                    hole, the mingling ofthe two lines of thought are simply
                    unavoidable. For in the event of a strong vacuum fluctuation
                    in such a region, in such circumstances so that one of the
                    pairs produced in that phenomena is captured by the black
                    hole and the other is not, we are left with the recourse
                    that the black hole itself must "radiate" in some capacity-
                    though the particulars of such radiation are open to a
                    certain level of interpretation. However, the reader can
                    begin to see now the potential links between this field and
                    quantum gravity (and we will see links to quantum
                    information theory too- all in due time!)
                </p>

                <p>
                    Speaking, conveniently, of particulars, one of the first 
                    things that needs to be covered in a ground-up approach is 
                    elementary thermal physics. So in the spirit of walking 
                    before we run, lets take some time to cover the necessities 
                    in that field.
                </p>

                <h5 id="zero"> Section 1.2 - The Zeroeth Law </h5>

                <p>
                    The most important concept to discuss in this article is 
                    entropy- we will see later that there are strong parallels
                    between thermodynamic entropy and informational entropy that
                    motivate much of the work in the field. However,before we 
                    touch on this idea, we should cover even more elementary
                    material: the zeroeth and first laws of thermodynamics. 
                </p>

                <p>
                    Like all sane people, we're going to index at zero and start
                    with that law. The zeroeth law of thermodynamics is fairly
                    intuitive.
                </p>
                    
                <p>
                    <b>ZEROETH LAW:</b>If two thermal systems are
                    in thermal equilibrium with a third system, then they
                    are in thermal equilibrium with each other. 
                </p>

                <p>
                    For the mathematicians in the room, this is essentially a
                    statement asserting that thermal equilibrium obeys the 
                    transitive property. Further, we consider it obvious that 
                    the notion of thermal equilibrium is reflexive and 
                    symmetric- which makes thermal equilibrium an equivalence
                    relation. We call elements belonging to a given partition of
                    this equivalence class "isotherms".
                </p>
                    
                 <p>
                    That's all well and good, but we're still under the hood of
                    some jargon- what exactly is thermal equilibrium? In order
                    to answer that, I'm going to have to introduce some more
                    jargon, but if you can stick through it, you'll understand
                    soon enough.
                </p>

                <p>
                    We say systems are in thermal equilibrium if, on average, no
                    thermal energy is being transfered between them. We say on
                    average because in the real world, quantum mechanics plays
                    a part in physics and will in fact result in minor changes
                    to levels of thermal energy at an instant in time- but in
                    the long run these average to zero. We've traded one term
                    for another: we now need to ask what thermal energy is.
                </p>
                
                <p>
                    I'm so glad you asked.
                </p>

                <h5 id = "energy">Section 1.3 - Heat, Work, and Energy: The First Law</h5>

                <p>
                    Thermal energy is difficult to explain directly because
                    energy itself bears some explanation- the energy of a system
                    may be thought of as that body's capacity to do work: to
                    apply some force perpendicular to some distance traveled in
                    a given direction. Energy itself takes form in, broadly
                    speaking, two ways: kinetic, the energy due to the motion of
                    a body represented for our purposes by $T$ and given by 
                    some variation of the quadratic form $mv^2$ (we also see 
                    rotational and vibrational modes of kinetic energy- they too
                    have a quadratic form. More on them later.) We also have 
                    potential energies, which are dependent on a body's relation
                    to other bodies in a positional sense. We denote potential 
                    energy with $V$ and write that the total energy, $U$, is 
                    given by the equation $$U = T + V.$$ 
                    
                    In thermodynamics, we have two ways for changes in energy to
                    express themselves: in the forms of work, being energy 
                    transmitted actively and quantifiably via some external 
                    mechanism- the cannonical example being the compressive 
                    action of a piston on a cylindrical container. We often 
                    denote work with the symbol $W$. 
                </p>

                <p>
                    We also have a notion of spontaneous energy changes is
                    systems, which we call "heat", and represent with the 
                    symbol $Q$. This spontaneity is key- the energy transfer
                    not arise out of the external action of some mechanism, or
                    we would have to classify it as work. There are three
                    well-described ways that heat may be transferred: through
                    <a href ="https://en.wikipedia.org/wiki/Thermal_conduction", target="_blank">conduction</a>,
                    <a href ="https://en.wikipedia.org/wiki/Convective_heat_transfer", target="_blank">convection</a>,
                    and  <a href ="https://en.wikipedia.org/wiki/Thermal_radiation", target="_blank">radiation</a>.
                    The particulars of each process are not material at the
                    moment, and I will leave it to the curious to familiarize
                    themselves with the notions if they are not conversant
                    already. They bare mentioning now because we will need some
                    understanding of radiative processes later. 
                </p>

                <p>
                    As we might say the total energy in a dynamical system is
                    $U = T + V$, we say that the total change in energy in a 
                    thermal system is give by $\Delta U = W + Q$. This is 
                    essentially a statement of the first law of thermodynamics,
                    but we might as well make that explicit.
                </p>
                
                <p>
                    <b>FIRST LAW:</b> The total energy of an isolated system 
                    (i.e. any system free from outside interaction) is conserved
                    and the changes in energy of a closed system (i.e. one free
                    from exchanges in matter but free to exchange energy) are 
                    given by $$\Delta U = W + Q$$ following the IUPAC conention.
                </p>

                <p>
                    At this point, it is necessary to introduce the idea of 
                    temperature and its relationship to thermal energy. To do
                    that, I want to introduce yet another element into the mix.
                </p>
                
                <h5 id="ideal">1.4 - The Ideal Gas Law

                <p>
                    The ideal gas law is a result that predicts certain 
                    properties of gasses in "ideal" conditions- in which the 
                    particles in the gas are moving fairly quickly (but not too
                    quickly) and are in general not close enough together to
                    be succeptible to interactions with the other particles. 
                    It was first concieved of as an amalgamation of many results
                    chemists had found in the early 19th century, but may also
                    be thought of as a result 
                    <a href="/articles/appendices/bht-a1/">derived</a> from the 
                    <a href="https://en.wikipedia.org/wiki/Equipartition_theorem", target="_blank">equipartition theorem</a>.
                    One's perspective relies on where they start and what they
                    take to be a first principle; I was taught this way and so
                    that's the way I'm going to present the material. 
                </p>

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