Digitization equipment spread.

From the front matter of Catalogue of Stars by Caroline Lucretia Herschel and John Flamsteed (1798). Original from the Bavarian State Library. Digitized January 14, 2011.

Marbled endpapers, with black-and-red optimization glitch.

From the back matter of A Select Collection of the Best Modern English Plays, v.4 (1750). Original from the Bavarian State Library. Digitized September 28, 2011.

VENTO AUREO CHARACTER OPTIMIZATION

PART 7/7!: LIGHT NOVELS!

what, you thought I wasn’t gonna draw them?? ALSO THIS IS MY 999th POST!

thank you to golden heart golden ring to just not really giving us full character or stand refs for every character thus allowing me artistic liberty beyond belief

Maximum Entropy Distributions

Entropy is an important topic in many fields; it has very well known uses in statistical mechanics,thermodynamics, and information theory. The classical formula for entropy is Σ_i(p_i log p_i), where p=p(x) is a probability density function describing the likelihood of a possible microstate of the system, i, being assumed. But what is this probability density function? How must the likelihood of states be configured so that we observe the appropriate macrostates?

In accordance with the second law of thermodynamics, we wish for the entropy to be maximized. If we take the entropy in the limit of large N, we can treat it with calculus as S[φ]=∫dx φ ln φ. Here, S is called a functional (which is, essentially, a function that takes another function as its argument). How can we maximize S? We will proceed using the methods of calculus of variationsandLagrange multipliers.

First we introduce three constraints. We require normalization, so that ∫dx φ = 1. This is a condition that any probability distribution must satisfy, so that the total probability over the domain of possible values is unity (since we’re asking for the probability of any possible event occurring). We require symmetry, so that the expected value of x is zero (it is equally likely to be in microstates to the left of the mean as it is to be in microstates to the right — note that this derivation is treating the one-dimensional case for simplicity). Then our constraint is ∫dx x·φ = 0. Finally, we will explicitly declare our variance to be σ², so that ∫dx x²·φ = σ².

Using Lagrange multipliers, we will instead maximize the augmented functional S[φ]=∫(φ ln φ + λ₀φ + λ₁xφ + λ₂x²φ dx). Here, the integrand is just the sum of the integrands above, adjusted by Lagrange multipliers λ_k for which we’ll be solving.

Applying the Euler-Lagrange equations and solving for φ gives φ = 1/exp(1+λ₀+xλ₁+x²λ₂). From here, our symmetry condition forces λ₁=0, and evaluating the other integral conditions gives our other λ’s such that q = (1/2πσ²)^½·exp(-x² / 2σ²), which is just the Normal (or Gaussian) distribution with mean 0 and variance σ². This remarkable distribution appears in many descriptions of nature, in no small part due to the Central Limit Theorem.

Get to know Marc Schwalbach, computational scientist

1) What do you do?

I specialize in the field of computational sciences / scientific computing. This is essentially a combination of computer science, maths, and engineering that allows us to use computers to simulate physical phenomena, such as the flow of fluids around an airplane or weather forecasting. These computer simulations serve as a virtual lab with which we can analyze different designs and scenarios even before the first prototype is manufactured for physical experiments.

For example, in my PhD I combined both fluid and structural simulations to optimize the shape of a radial turbine to maximize its aerodynamic efficiency while ensuring its structural integrity.

2) Where do you work?

I currently work as a research engineer at CFD-Berlin in Germany, where most projects involve computational fluid dynamics and aeroacoustic optimization.

3) Tell us about the photos!

[Left:]My COVID-times PhD defense pose which was conducted via video call earlier this year.

[Right:] At a project meeting in Glasgow with our international partners in 2018

4) Tell us about your academic career path so far. 

After graduating from high school in the Philippines, I moved to Germany where I first attended a preparatory “Studienkolleg” that allows students with foreign high school degrees to apply for university. 

I received my B.Sc. and M.Sc. in Computational Engineering Science from RWTH Aachen University where I began specializing in computational  fluid dynamics and optimization. During this time, I got to spend a few months as a research intern at UBC in Canada with a DAAD RISE scholarship.

I then moved to the von Karman Institute in Belgium where I conducted my PhD research as a Marie Curie fellow and received my doctoral degree in Computer Science from the TU Kaiserslautern in Germany.

5) Anything else you’d like to share

This site builds a strong sense of community among Pinoy Scientists and it’s encouraging to see that many of us abroad keep our roots in mind. I hope youngsters who are thinking about a scientific or technical career see their future selves in some of the stories posted here and that more Pinoy Scientists take the initiative to contribute. If anyone would like to get in touch, please don’t hesitate: marcschwalbach.com.