Astrophysics, with its roots in ancient stellar observatories, sought to unravel the enigmas of the celestial phenomena that adorned the night sky. As telescopes evolved and space exploration burgeoned, the demand for a robust mathematical framework became increasingly apparent, leading to the integration of calculus.
Solving the Mystery of Bose-Einstein Condensates
Millions of atoms can synchronize into one quantum state near absolute zero, creating a rare fifth state of matter.
University of Copenhagen professor Søren Fournais studies Bose-Einstein condensates, where massive groups of atoms behave identically in a synchronized quantum state.
Bose-Einstein Condensates
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Predicted by Indian physicist Satyendra Nath Bose and later expanded by Einstein in the 1920s.
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First successfully created in a laboratory in 1995.
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Considered a unique fifth state of matter.
Researchers still struggle to mathematically explain how millions of interacting particles transition into the same quantum state.
The Mathematical Challenge
Bose-Einstein condensation is a phase transition similar to water freezing into ice, but proving it mathematically remains extremely difficult.
Fournais hopes to better understand the equations behind these quantum systems through theoretical mathematics rather than computer simulations.
i) If n = 1, 1A + 1B + 1C = 1 (that is, first term is 1)
ii) If n = 2, 4A + 2B + 1C = -2 (that is, second term is -2)
iii) If n = 3, 9A + 3B + 1C = -1 (that is, third term is -1)
The first finite differences are determined by subtracting the consecutive terms in original sequence. That is, take -2-1=-3, -1-(-2)=1, 4-(-1)=5, 13-4=9, 26-13=13, and so on.
The second differences are determined by subtracting the consecutive first differences. When the second differences are all similar, then the pattern is quadratic. Keep in mind that you can determine a quadratic model by taking the equation y = ax2 + bx + c with three points. Then resolve the system of equations which results. The analogy here is that you can find out an = an2 + bn + c by replacing in three terms in the sequence and their corresponding place in the sequence for n. Then resolve the system of linear equations.
Finite Differences illustration:
