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Cell["3.029 Spring 2022\[LineSeparator]Lecture 06 - 02/16/2022", "Subtitle",
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Last lecture, we introduced the concept of Legendre transforms as a way of \
\[OpenCurlyDoubleQuote]changing variables\[CloseCurlyDoubleQuote] from one \
thermodynamic potential to another\
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Define a new thermodynamic potential,                    \[CapitalPhi] = U - \
Y X\
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Take its total differential,                                               d\
\[CapitalPhi] = dU - Y dX - X dY\
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Use the fundamental thermodynamic relation,  dU = T dS - P dV\
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As a reminder, the key thermodynamic potentials we defined this way are:\
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Internal Energy,\t                               dU(S,V) = T dS - P dV\
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 "Entropy,                                               dS(U,V) = ",
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Enthalpy,                                             dH(S,P) = T dS + V dP\
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Helmholtz Free Energy,                 dA(T,V) = -S dT - P dV\
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Gibbs Free Energy,                           dG(T,P) = -S dT + V dP\
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 "This procedure is not restricted to named potentials. ",
 "If you have other work terms in your potential, \[LineSeparator]\
\[LineSeparator]e.g. electrical work in an dielectric medium, \
\[ScriptCapitalE] \[ScriptCapitalP], where \[ScriptCapitalE] is the \
(intensive) internal electric field, and \[ScriptCapitalP] is the (extensive) \
total polarization,\[LineSeparator]then the fundamental thermodynamic \
relation would read: d U = T dS - P dV + \[ScriptCapitalE] d\[ScriptCapitalP] \
(remember, the variations are wrt extensive variables),\[LineSeparator]\
\[LineSeparator]and we want to instead express this in terms of the internal \
electric field, we would define:\[LineSeparator]\[LineSeparator]  \
\[CapitalPhi]  = U - \[ScriptCapitalE] \[ScriptCapitalP]\[LineSeparator]d\
\[CapitalPhi] = d U - \[ScriptCapitalE] d\[ScriptCapitalP]  - \
\[ScriptCapitalP] d\[ScriptCapitalE]\[LineSeparator]d\[CapitalPhi] = T dS - \
\[ScriptCapitalP] d\[ScriptCapitalE]"
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This is largely all you will need in your thermodynamics and statistical \
mechanics courses\
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But in this lecture, we will go a step further, and provide an intuitive \
geometrical picture of what the Legendre transform does\
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The Legendre transform is defined for general functions, but it is \
particularly useful in looking at convex functions\
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The definition we used above for convexity can be generalized to higher \
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Instead, we will consider what \[OpenCurlyDoubleQuote]non-convexity\
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Let\[CloseCurlyQuote]s consider the a 2D surface embedded in 3D dimensions \
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Cell["\<\
\[OpenCurlyDoubleQuote]Non-convexity\[CloseCurlyDoubleQuote] mean that:\
\[LineSeparator]for any point {x,y,z} on the surface, there are other points \
that can be connected with a hyperplane (the higher-dimensional \
generalization of a line) and have part of the plane be \
\[OpenCurlyDoubleQuote]outside\[CloseCurlyDoubleQuote] the surface. \
\>", "Item",
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Cell["We can \[OpenCurlyDoubleQuote]convexify\[CloseCurlyDoubleQuote] regions \
by constructing a convex hull ", "Item",
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Cell["\<\
We will revisit the concept of a Convex Hull many times during our course!\
\>", "Subitem",
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For now, we note that it\[CloseCurlyQuote]s a convex envelope which includes \
all points on our shape, and satisfies the convexity criterion above\
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Cell["\<\
Such \[OpenCurlyDoubleQuote]convexification\[CloseCurlyDoubleQuote] is what \
one would use if the molar internal energy were a function of two molar \
variables\
\>", "Item",
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Cell["\<\
Another very important application of convex functions in Materials Science, \
is in constructing equilibrium crystal shapes based on the surface energy per \
unit are of different crystal orientations\
\>", "Item",
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Cell["We will investigate this later in the course!", "Subitem",
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Cell["Geometric Interpretation", "Section",
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Cell["\<\
Now that we understand convex functions, let\[CloseCurlyQuote]s go back to \
our Legendre transform and start by looking at a convex function in a single \
variable\
\>", "Item",
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Cell["\<\
We can use the built-in FunctionConvexity to ensure the function is indeed \
convex\
\>", "Item",
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Cell["\<\
This is because Log of a negative number is complex-valued.\[LineSeparator]We \
will restrict our range between 0-1 for now\
\>", "Subitem",
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Cell["Of-course an easier way would be to just plot it!", "Item",
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Of-course the Hamiltonian \[LeftRightArrow] Lagrangian transformation is also \
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First, we define the velocity as the derivative of our Hamiltonian\
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Aha, here\[CloseCurlyQuote]s the culprit! \[LineSeparator]Two branches - we\
\[CloseCurlyQuote]ll pick the positive one\
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