78. Jorge Do Marco, R.; Giordano, S. Thermodynamics of Extra-Toughness and Hidden-Length in Polymeric Materials with Sacrificial Bonds. *Appl. Mech. 2022*, **3**, 935–955. [CrossRef]

79. Cannizzo, A.; Giordano, S. Thermal effects on fracture and the brittle-to-ductile transition. *Phys. Rev. E* **2023**, *107*, 035001. [CrossRef]

80. Doi, M. *Introduction to Polymer Physics*; Oxford University Press: Oxford, NY, USA, 1996

81. Giordano, S. Helmholtz and Gibbs ensembles, thermodynamic limit and bistability in polymer lattice models. *Contin. Mech. Thermodyn.* **2018**, *30*, 459–483. [CrossRef]

82. Buche, M.R.; Silberstein, M.N. Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble. *Phys. Rev. E* **2020**, *102*, 012501. [CrossRef] [PubMed]

83. Buche, M.R.; Silberstein, M.N.; Grutzik, S.J. Freely jointed chain models with extensible links. *Phys. Rev. E* **2022**, *106*, 024502. [CrossRef] [PubMed]

84. Cleri, F. Microscopic mechanics of biomolecules in living cells. *Sci. Model. Simul.* **2008**, *15*, 339–362. [CrossRef]

85. Kumar, S.; Li, M.S. Biomolecules under mechanical force. *Phys. Rep.* **2010**, *486*, 1–74. [CrossRef]

86. Mandal, S.S. Force Spectroscopy on Single Molecules of Life. *ACS Omega* **2020**, *5*, 11271–11278. [CrossRef] [PubMed]

87. Neuman, K.C.; Nagy, A. Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. *Nat. Methods* **2008**, *5*, 491–505. [CrossRef] [PubMed]

88. Ritort, F. Single-molecule experiments in biological physics: Methods and applications. *J. Phys. Condens. Matter* **2006**, *18*, R531. [CrossRef] [PubMed]

89. Fisher, T.E.; Oberhauser, A.F.; Carrion-Vazquez, M.; Marszalek, P.E.; Fernandez, J.M. The study of protein mechanics with the atomic force microscope. *Trends Biochem. Sci.* **1999**, *24*, 379–384. [CrossRef] [PubMed]

90. Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H.E. Single Molecule Force Spectroscopy on Polysaccharides by Atomic Force Microscopy. *Science* **1997**, *275*, 1295–1297. [CrossRef] [PubMed]

91. Rief, M.; Gautel, M.; Oesterhelt, F.; Fernandez, J.M.; Gaub, H.E. Reversible Unfolding of Individual Titin Immunoglobulin Domains by AFM. *Science* **1997**, *276*, 1109–1112. [CrossRef]

92. Rief, M.; Fernandez, J.M.; Gaub, H.E. Elastically Coupled Two-Level Systems as a Model for Biopolymer Extensibility. *Phys. Rev. Lett.* **1998**, *81*, 4764–4767. [CrossRef]

93. Rief, M.; Pascual, J.; Saraste, M.; Gaub, H.E. Single molecule force spectroscopy of spectrin repeats: Low unfolding forces in helix bundles. *J. Mol. Biol.* **1999**, *286*, 553–561. [CrossRef]

94. Cocco, S., Yan, J., Léger, J.-F., Chatenay, D.; Marko, J.F. Overstretching and force-driven strand separation of double-helix DNA. *Phys. Rev. E* **2004**, *70*, 011910. [CrossRef]

95. Manghi, M.; Destainville, N.; Palmeri, J. Mesoscopic models for DNA stretching under force: New results and comparison with experiments. *Eur. Phys. J. E* **2012**, *35*, 110. [CrossRef] [PubMed]

96. Marin-Gonzalez, A.; Vilhena, J.G.; Perez, R.; Moreno-Herrero, F. Understanding the mechanical response of double-stranded DNA and RNA under constant stretching forces using all-atom molecular dynamics. *Proc. Natl. Acad. Sci. USA* **2017**, *114*, 7049–7054. [CrossRef] [PubMed]

97. Florio, G.; Puglisi, G. A predictive model for the thermomechanical melting transition of double stranded DNA. *Acta Biomater.* **2023**, *157*, 225–235. [CrossRef] [PubMed]

98. Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: Fifty years after Kramers. *Rev. Mod. Phys.* **1990**, *62*, 251–341. [CrossRef]

99. Kramers, H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. *Physica* **1940**, *7*, 284–304. [CrossRef]

100. Berglund, N. Kramers' law: Validity, derivations and generalisations. *Markov Process. Relat. Fields* **2013**, *19*, 459–490. [CrossRef]

101. Pollak, E.; Miret-Artés, S. Recent Developments in Kramers' Theory of Reaction Rates. *ChemPhysChem* **2023**, *24*, e202300272. [CrossRef] [PubMed]

102. Coffey, W.T.; Kalmykov, Y.P. *The Langevin Equation*; World Scientific Publishing: Singapore, 2012.

103. Giordano, S.; Cleri, F.; Blossey, R. On the One-Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells. *Ann. Phys.* **2023**, *535*, 2300294. [CrossRef]

104. Rico, F.; Gonzalez, L.; Casuso, I.; Puig-Vidal, M.; Scheuring, S. High-speed force spectroscopy unfolds titin at the velocity of molecular dynamics simulations. *Science* **2013**, *342*, 741–743. [CrossRef]

105. Eghiaian, F.; Rico, F.; Colom, A.; Casuso, I.; Scheuring, S. High-speed atomic force microscopy: Imaging and force spectroscopy. *FEBS Lett.* **2014**, *588*, 3631–3638. [CrossRef]

106. Manca, F.; Giordano, S.; Palla, P.L.; Zucca, R.; Cleri, F.; Colombo, L. Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles. *J. Chem. Phys.* **2012**, *136*, 154906. [CrossRef] [PubMed]

107. Manca, F.; Giordano, S.; Palla, P.L.; Cleri, F.; Colombo, L. Response to "Comment on 'Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles". *J. Chem. Phys.* **2013**, *138*, 157102. [CrossRef] [PubMed]

108. Manca, F.; Giordano, S.; Palla, P.L.; Cleri, F. On the equivalence of thermodynamics ensembles for flexible polymer chains. *Phys. A Stat. Mech. Appl.* **2014**, *395*, 154–170. [CrossRef]

109. Dimitrov, D.I.; Klushin, L.I.; Skvortsov, A.; Milchev, A.; Binder, K. The escape transition of a polymer: A unique case of non-equivalence between statistical ensembles. *Eur. Phys. J. E* **2009**, *29*, 9–25. [CrossRef] [PubMed]

110. Winkler, R.G. Equivalence of statistical ensembles in stretching single flexible polymers. *Soft Matter* **2010**, *6*, 6183–6191. [CrossRef]
78. Jorge Do Marco, R.; Giordano, S. Thermodynamics of Extra-Toughness and Hidden-Length in Polymeric Materials with Sacrificial Bonds. *Appl. Mech. 2022*, **3**, 935–955. [CrossRef]

79. Cannizzo, A.; Giordano, S. Thermal effects on fracture and the brittle-to-ductile transition. *Phys. Rev. E* **2023**, *107*, 035001. [CrossRef]

80. Doi, M. *Introduction to Polymer Physics*; Oxford University Press: Oxford, NY, USA, 1996

81. Giordano, S. Helmholtz and Gibbs ensembles, thermodynamic limit and bistability in polymer lattice models. *Contin. Mech. Thermodyn.* **2018**, *30*, 459–483. [CrossRef]

82. Buche, M.R.; Silberstein, M.N. Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble. *Phys. Rev. E* **2020**, *102*, 012501. [CrossRef] [PubMed]

83. Buche, M.R.; Silberstein, M.N.; Grutzik, S.J. Freely jointed chain models with extensible links. *Phys. Rev. E* **2022**, *106*, 024502. [CrossRef] [PubMed]

84. Cleri, F. Microscopic mechanics of biomolecules in living cells. *Sci. Model. Simul.* **2008**, *15*, 339–362. [CrossRef]

85. Kumar, S.; Li, M.S. Biomolecules under mechanical force. *Phys. Rep.* **2010**, *486*, 1–74. [CrossRef]

86. Mandal, S.S. Force Spectroscopy on Single Molecules of Life. *ACS Omega* **2020**, *5*, 11271–11278. [CrossRef] [PubMed]

87. Neuman, K.C.; Nagy, A. Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. *Nat. Methods* **2008**, *5*, 491–505. [CrossRef] [PubMed]

88. Ritort, F. Single-molecule experiments in biological physics: Methods and applications. *J. Phys. Condens. Matter* **2006**, *18*, R531. [CrossRef] [PubMed]

89. Fisher, T.E.; Oberhauser, A.F.; Carrion-Vazquez, M.; Marszalek, P.E.; Fernandez, J.M. The study of protein mechanics with the atomic force microscope. *Trends Biochem. Sci.* **1999**, *24*, 379–384. [CrossRef] [PubMed]

90. Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H.E. Single Molecule Force Spectroscopy on Polysaccharides by Atomic Force Microscopy. *Science* **1997**, *275*, 1295–1297. [CrossRef] [PubMed]

91. Rief, M.; Gautel, M.; Oesterhelt, F.; Fernandez, J.M.; Gaub, H.E. Reversible Unfolding of Individual Titin Immunoglobulin Domains by AFM. *Science* **1997**, *276*, 1109–1112. [CrossRef]

92. Rief, M.; Fernandez, J.M.; Gaub, H.E. Elastically Coupled Two-Level Systems as a Model for Biopolymer Extensibility. *Phys. Rev. Lett.* **1998**, *81*, 4764–4767. [CrossRef]

93. Rief, M.; Pascual, J.; Saraste, M.; Gaub, H.E. Single molecule force spectroscopy of spectrin repeats: Low unfolding forces in helix bundles. *J. Mol. Biol.* **1999**, *286*, 553–561. [CrossRef]

94. Cocco, S., Yan, J., Léger, J.-F., Chatenay, D.; Marko, J.F. Overstretching and force-driven strand separation of double-helix DNA. *Phys. Rev. E* **2004**, *70*, 011910. [CrossRef]

95. Manghi, M.; Destainville, N.; Palmeri, J. Mesoscopic models for DNA stretching under force: New results and comparison with experiments. *Eur. Phys. J. E* **2012**, *35*, 110. [CrossRef] [PubMed]

96. Marin-Gonzalez, A.; Vilhena, J.G.; Perez, R.; Moreno-Herrero, F. Understanding the mechanical response of double-stranded DNA and RNA under constant stretching forces using all-atom molecular dynamics. *Proc. Natl. Acad. Sci. USA* **2017**, *114*, 7049–7054. [CrossRef] [PubMed]

97. Florio, G.; Puglisi, G. A predictive model for the thermomechanical melting transition of double stranded DNA. *Acta Biomater.* **2023**, *157*, 225–235. [CrossRef] [PubMed]

98. Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: Fifty years after Kramers. *Rev. Mod. Phys.* **1990**, *62*, 251–341. [CrossRef]

99. Kramers, H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. *Physica* **1940**, *7*, 284–304. [CrossRef]

100. Berglund, N. Kramers' law: Validity, derivations and generalisations. *Markov Process. Relat. Fields* **2013**, *19*, 459–490. [CrossRef]

101. Pollak, E.; Miret-Artés, S. Recent Developments in Kramers' Theory of Reaction Rates. *ChemPhysChem* **2023**, *24*, e202300272. [CrossRef] [PubMed]

102. Coffey, W.T.; Kalmykov, Y.P. *The Langevin Equation*; World Scientific Publishing: Singapore, 2012.

103. Giordano, S.; Cleri, F.; Blossey, R. On the One-Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells. *Ann. Phys.* **2023**, *535*, 2300294. [CrossRef]

104. Rico, F.; Gonzalez, L.; Casuso, I.; Puig-Vidal, M.; Scheuring, S. High-speed force spectroscopy unfolds titin at the velocity of molecular dynamics simulations. *Science* **2013**, *342*, 741–743. [CrossRef]

105. Eghiaian, F.; Rico, F.; Colom, A.; Casuso, I.; Scheuring, S. High-speed atomic force microscopy: Imaging and force spectroscopy. *FEBS Lett.* **2014**, *588*, 3631–3638. [CrossRef]

106. Manca, F.; Giordano, S.; Palla, P.L.; Zucca, R.; Cleri, F.; Colombo, L. Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles. *J. Chem. Phys.* **2012**, *136*, 154906. [CrossRef] [PubMed]

107. Manca, F.; Giordano, S.; Palla, P.L.; Cleri, F.; Colombo, L. Response to "Comment on 'Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles". *J. Chem. Phys.* **2013**, *138*, 157102. [CrossRef] [PubMed]

108. Manca, F.; Giordano, S.; Palla, P.L.; Cleri, F. On the equivalence of thermodynamics ensembles for flexible polymer chains. *Phys. A Stat. Mech. Appl.* **2014**, *395*, 154–170. [CrossRef]

109. Dimitrov, D.I.; Klushin, L.I.; Skvortsov, A.; Milchev, A.; Binder, K. The escape transition of a polymer: A unique case of non-equivalence between statistical ensembles. *Eur. Phys. J. E* **2009**, *29*, 9–25. [CrossRef] [PubMed]

110. Winkler, R.G. Equivalence of statistical ensembles in stretching single flexible polymers. *Soft Matter* **2010**, *6*, 6183–6191. [CrossRef]
111. Dutta, S.; Benetatos, P. Inequivalence of fixed-force and fixed-extension statistical ensembles for a flexible polymer tethered to a planar substrate. *Soft Matter* **2018**, *14*, 6857–6866. [CrossRef] [PubMed]

112. Dutta, S.; Benetatos, P. Statistical ensemble inequivalence for flexible polymers under confinement in various geometries. *Soft Matter* **2019**, *16*, 2114–2127. [CrossRef]

113. Noh, G.; Benetatos, P. Tensile elasticity of a freely jointed chain with reversible hinges. *Soft Matter* **2021**, *17*, 3333–3345. [CrossRef]

114. Gammaitoni, L.; Marchesoni, F.; Menichella-Saetta, E.; Santucci, S. Stochastic Resonance in Bistable Systems. *Phys. Rev. Lett.* **1989**, *62*, 349–352. [CrossRef]

115. Gammaitoni, L.; Hanggi, P.; Jung, P.; Marchesoni, F. Stochastic resonance. *Rev. Mod. Phys.* **1998**, *70*, 223–287. [CrossRef]

116. Kima, W.K.; Hyeon, C.; Sung, W. Weak temporal signals can synchronize and acceleratethe transition dynamics of biopolymers under tension. *Proc. Natl. Acad. Sci. USA* **2012**, *109*, 14410–14415. [CrossRef] [PubMed]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
111. Dutta, S.; Benetatos, P. Inequivalence of fixed-force and fixed-extension statistical ensembles for a flexible polymer tethered to a planar substrate. *Soft Matter* **2018**, *14*, 6857–6866. [CrossRef] [PubMed]

112. Dutta, S.; Benetatos, P. Statistical ensemble inequivalence for flexible polymers under confinement in various geometries. *Soft Matter* **2019**, *16*, 2114–2127. [CrossRef]

113. Noh, G.; Benetatos, P. Tensile elasticity of a freely jointed chain with reversible hinges. *Soft Matter* **2021**, *17*, 3333–3345. [CrossRef]

114. Gammaitoni, L.; Marchesoni, F.; Menichella-Saetta, E.; Santucci, S. Stochastic Resonance in Bistable Systems. *Phys. Rev. Lett.* **1989**, *62*, 349–352. [CrossRef]

115. Gammaitoni, L.; Hanggi, P.; Jung, P.; Marchesoni, F. Stochastic resonance. *Rev. Mod. Phys.* **1998**, *70*, 223–287. [CrossRef]

116. Kima, W.K.; Hyeon, C.; Sung, W. Weak temporal signals can synchronize and acceleratethe transition dynamics of biopolymers under tension. *Proc. Natl. Acad. Sci. USA* **2012**, *109*, 14410–14415. [CrossRef] [PubMed]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.