Presentation file for "Thermal stability originates the vanishing of the specific heat at absolute zero"

Thermal stability originates the vanishing of the specific heats at absolute zero Physica Scripta (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 Jos´e Mar´ıa Mart´ın Olalla (0000-0002-3750-9113) Departamento de F´ısica de la Materia Condensada (03yxnpp24) 2 December 2025 https://youtu.be/V5jFDAnH3Go Contents 1Problem statement 2Specific heat, curvature and stability 3Close to absolute zero 4Conclusion Problem statement Energy, U First law Temperature, T > 0 Entropy, S Second law Cv=T∂S ∂T v >0 Definition and domain Equilibrium conditions Stability conditions Fundamental relation, U(S, V, N) Uss =∂2U ∂S2v =∂T ∂S v =T Cv >0 Regular stability Uss =∂2U ∂S2v =∂T ∂S v =T Cv = 0 Critical stability https://www.youtube.com/watch?v=VGxWFWiQQRc Non-zero temperature evidence lim T→0+∂S ∂V t = 0 lim T→0+T∂S ∂T v = 0 “Third law” Realm of absolute zero Physica Scripta 2025 NEW Universidad de Sevilla. JM Mart´ın-Olalla Phys. Scr. (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 by L A T E X 2ε-beamer,PGF/TikZ,OBS,Gemini 1/8 Contents 1Problem statement 2Specific heat, curvature and stability 3Close to absolute zero 4Conclusion Increasing strictly convex ∪energy is locally stable s−δs ss+δs u(s−δs) u(s) u(s+δs) osculating circle κ∝uss =∂2u ∂s2v =∂T ∂s v =T cv v= cte s u 1 2u(s+δs) + u(s−δs)> u(s)(1) Inhomogeneity is abhorred because it yields more energy for the same entropy. Sufficient condition: uss =∂2u ∂s2v =∂T ∂s v =T cv >0.(2) If T > 0, then cv>0. Universidad de Sevilla. JM Mart´ın-Olalla Phys. Scr. (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 by L A T E X 2ε-beamer,PGF/TikZ,OBS,Gemini 2/8 Loss of curvature causes inestability (critical fluctuations) sc uc s u1 2u(s+δs) + u(s−δs)∼u(s)(3) Inhomogeneities can grow inmensely due to the “soft mode”. https://www. youtube.com/watch?v=VGxWFWiQQRc Necessary condition: uss =∂2u ∂s2v =∂T ∂s v =T cv = 0.(4) If T > 0, then cv→ ∞. Universidad de Sevilla. JM Mart´ın-Olalla Phys. Scr. (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 by L A T E X 2ε-beamer,PGF/TikZ,OBS,Gemini 3/8 Contents 1Problem statement 2Specific heat, curvature and stability 3Close to absolute zero 4Conclusion How can we land at T= 0? (n={1.1,1.3,1.5,1.75}) T=∂u ∂s v →0+ lim T→0+uss = lim T→0+ T cv ? ≥0 u(s) = sn, s ≥0, n > 1,(5) T(s) = nsn−1,(6) uss =n(n−1)sn−2,(7) cv∝T1/(n−1).(8) 0 1 0 1 T= 0 m= 1 n= 2 s u Universidad de Sevilla. JM Mart´ın-Olalla Phys. Scr. (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 by L A T E X 2ε-beamer,PGF/TikZ,OBS,Gemini 4/8 Evidence and models concur lim T→0+uss = lim T→0+ T cv ? ≥0 Cases 1lim T→0+uss >0(stable): cvvanishes as fast as Tor faster. 2lim T→0+uss = 0 (critical): cvvanishes slower than T. cvdoes not vanish at all. (asymptotic behaviour: T= 0 only when S→ −∞) 0 1 0 1 T= 0 m= 1 lim T→0+uss >0, stable lim T→0+uss = 0, critical electronic contribution crystalline solid two states Quantum critical point s u Universidad de Sevilla. JM Mart´ın-Olalla Phys. Scr. (2025) 100 125206 doi:10.1088/1402-4896/ae22a5 by L A T E X 2ε-beamer,PGF/TikZ,OBS,Gemini 5/8