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I'm getting back into information geometry, which is the geometry of the space of probability distributions, studied using tools from information theory. Now I'm fascinated by something new: how symplectic geometry and contact geometry show up in information geometry. But before I say anything about this, let me say a bit about how they show up in thermodynamics. This is more widely discussed, and it's a good starting point.
Symplectic geometry was born as the geometry of phase space in classical mechanics: that is, the space of possible positions and momenta of a classical system. The simplest example of a symplectic manifold is the vector space \(\mathbb{R}^{2n},\) with n position coordinates \(q_i\) and n momentum coordinates \(p_i\).
It turns out that symplectic manifolds are always even-dimensional, because we can always cover them with coordinate charts that look like \(\mathbb{R}^{2n}\). When we change coordinates, it turns out that the splitting of coordinates into positions and momenta is somewhat arbitrary. For example, the position of a rock on a spring now may determine its momentum a while later, and vice versa. What's not arbitrary? It's the so-called 'symplectic structure':
$$ \omega = dp_1 \wedge dq_1 + \cdots + dp_n \wedge dq_n $$While far from obvious initially, we know by now that the symplectic structure is exactly what needs to be preserved under valid changes of coordinates in classical mechanics! In fact, we can develop the whole formalism of classical mechanics starting from a manifold with a symplectic structure.
Symplectic geometry also shows up in thermodynamics. In thermodynamics we can start with a system in equilibrium whose state is described by some variables \(q_1, \dots, q_n.\) Its entropy will be a function of these variables, say
$$ S = f(q_1, \dots, q_n) $$We can then take the partial derivatives of entropy and call them something:
$$ \displaystyle{ p_i = \frac{\partial f}{\partial q_i} } $$These new variables \(p_i\) are said to be 'conjugate' to the \(q_i,\) and they turn out to be very interesting. For example, if \(q_i\) is energy then \(p_i\) is 'coolness': the reciprocal of temperature. The coolness of a system is its change in entropy per change in energy.
Often the variables \(q_i\) are 'extensive': that is, you can measure them only by looking at your whole system and totaling up some quantity. Examples are energy and volume. Then the new variables \(p_i\) are 'intensive': that is, you can measure them at any one location in your system. Examples are coolness and pressure.
Now for a twist: sometimes we do not know the function \(f\) ahead of time. Then we cannot define the \(p_i\) as above. We're forced into a different approach where we treat them as independent quantities, at least until someone tells us what \(f\) is.
In this approach, we start with a space \(\mathbb{R}^{2n}\) having n coordinates called \(q_i\) and n coordinates called \(p_i.\) This is a symplectic manifold, with the symplectic struture \(\omega\) described earlier!
But what about the entropy? We don't yet know what it is as a function of the \(q_i,\) but we may still want to talk about it. So, we build a space \(\mathbb{R}^{2n+1}\) having one extra coordinate \(S\) in addition to the \(q_i\) and \(p_i.\) This new coordinate stands for entropy. And this new space has an important 1-form on it:
$$ \alpha = -dS + p_1 dq_i + \cdots + p_n dq_n $$This is called the 'contact 1-form'.
This makes \(\mathbb{R}^{2n+1}\) into an example of a 'contact manifold'. Contact geometry is the odd-dimensional partner of symplectic geometry. Just as symplectic manifolds are always even-dimensional, contact manifolds are always odd-dimensional.
What is the point of the contact 1-form? Well, suppose someone tells us the function \(f\) relating entropy to the coordinates \(q_i.\) Now we know that we want
$$ S = f $$ and also $$ \displaystyle{ p_i = \frac{\partial f}{\partial q_i} } $$So, we can impose these equations, which pick out a subset of \(\mathbb{R}^{2n+1}.\) You can check that this subset, say \(\Sigma,\) is an n-dimensional submanifold. But even better, the contact 1-form vanishes when restricted to this submanifold:
$$ \left.\alpha\right|_\Sigma = 0 $$Let's see why! Suppose \(x \in \Sigma\) and suppose \(v \in T_x \Sigma\) is a vector tangent to \(\Sigma\) at this point \(x\). It suffices to show
$$ \alpha(v) = 0 $$Using the definition of \(\alpha\) this equation says
$$ \displaystyle{ -dS(v) + \sum_i p_i dq_i(v) = 0 } $$But on the surface \(\Sigma\) we have
$$ S = f, \qquad \displaystyle{ p_i = \frac{\partial f}{\partial q_i} } $$So, the equation we're trying to show can be written as
$$ \displaystyle{ -df(v) + \sum_i \frac{\partial f}{\partial q_i} dq_i(v) = 0 }$$But this follows from
$$ \displaystyle{ df = \sum_i \frac{\partial f}{\partial q_i} dq_i } $$which holds because \(f\) is a function only of the coordinates \(q_i\).
So, any formula for entropy \(S = f(q_1, \dots, q_n)\) picks out a so-called 'Legendrian submanifold' of \(\mathbb{R}^{2n+1}:\) that is, an n-dimensional submanifold such that the contact 1-form vanishes when restricted to this submanifold. And the idea is that this submanifold tells you everything you need to know about a thermodynamic system.
Indeed, V. I. Arnol'd says this was implicitly known to the great founder of statistical mechanics, Josiah Willard Gibbs. Arnol'd calls \(\mathbb{R}^5\) with coordinates energy, entropy, temperature, pressure and volume the 'Gibbs manifold', and he proclaims:
Gibbs' thesis: substances are Legendrian submanifolds of the Gibbs manifold.
This is from here:
Now I want to say everything again, with a bit of extra detail, assuming more familiarity with manifolds. Above I was using \(\mathbb{R}^n\) with coordinates \(q_1, \dots, q_n\) to describe the extensive variables of a thermodynamic system. But let's be a bit more general and use any smooth n-dimensional manifold \(Q\). Even if \(Q\) is a vector space, this viewpoint is nice because it's manifestly coordinate-independent!
So: starting from \(Q\) we build the cotangent bundle \(T^\ast Q.\) A point in cotangent describes both extensive variables, namely \(q \in Q,\) and intensive variables, namely a cotangent vector \(p \in T^\ast_q Q\).
The manifold \(T^\ast Q\) has a 1-form \(\theta\) on it called the tautological 1-form. We can describe it as follows. Given a tangent vector \(v \in T_{(q,p)} T^\ast Q\) we have to say what \(\theta(v)\) is. Using the projection
$$ \pi \colon T^\ast Q \to Q $$we can project \(v\) down to a tangent vector \(d\pi(v)\) at the point \(q\). But the 1-form \(p\) eats tangent vectors at \(q\) and spits out numbers! So, we set
$$ \theta(v) = p(d\pi(v)) $$This is sort of mind-boggling at first, but it's worth pondering until it makes sense. It helps to work out what \(\theta\) looks like in local coordinates. Starting with any local coordinates \(q_i\) on an open set of \(Q,\) we get local coordinates \(q_i, p_i\) on the cotangent bundle of this open set in the usual way. On this open set you then get
$$ \theta = p_1 dq_1 + \cdots + p_n dq_n $$This is a standard calculation, which is really worth doing!
It follows that we can define a symplectic structure \(\omega\) by
$$ \omega = d \theta $$and get this formula in local coordinates:
$$ \omega = dp_1 \wedge dq_1 + \cdots + dp_n \wedge dq_n $$Now, suppose we choose a smooth function
$$ f \colon Q \to \mathbb{R} $$which describes the entropy. We get a 1-form \(df\), which we can think of as a map
$$ df \colon Q \to T^\ast Q $$assigning to each choice \(q\) of extensive variables the pair \((q,p)\) of extensive and intensive variables where
$$ p = df_q $$The image of the map \(df\) is a 'Lagrangian submanifold' of \(T^\ast Q:\) that is, an n-dimensional submanifold \(\Lambda\) such that
$$ \left.\omega\right|_{\Lambda} = 0 $$Lagrangian submanifolds are to symplectic geometry as Legendrian submanifolds are to contact geometry! What we're seeing here is that if Gibbs had preferred symplectic geometry, he could have described substances as Lagrangian submanifolds rather than Legendrian submanifolds. But this approach would only keep track of the derivatives of entropy, \(df\), not the actual value of the entropy function \(f\).
If we prefer to keep track of the actual value of \(f\) using contact geometry, we can do that. For this we add an extra dimension to \(T^\ast Q\) and form the manifold \(T^\ast Q \times \mathbb{R}.\) The extra dimension represents entropy, so we'll use \(S\) as our name for the coordinate on \(\mathbb{R}.\)
We can make \(T^\ast Q \times \mathbb{R}\) into a contact manifold with contact 1-form
$$ \alpha = -d S + \theta $$In local coordinates we get
$$ \alpha = -dS + p_1 dq_i + \cdots + p_n dq_n $$just as we had earlier. And just as before, if we choose a smooth function \(f \colon Q \to \mathbb{R}\) describing entropy, the subset
$$ \Sigma = \{(q,p,S) \in T^\ast Q \times \mathbb{R} : \; S = f(q), p = df_q \} $$is a Legendrian submanifold of \(T^\ast Q \times \mathbb{R}\).
Okay, this concludes my lightning review of symplectic and contact geometry in thermodynamics! Next time I'll talk about something a bit less well understood: how they show up in statistical mechanics.
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