Abstract
In this chapter we focus on the ensemble that describes a system in contact with a thermal reservoir and is subject to a constant pressure, i.e., the so called TP ensemble. Just like the canonical and microcanonical ensembles it accounts for conservative character of the thermrodynamic field, and allows for a general expression of the thermodynamic potential, i.e. the entropy. We study the ideal gas thermodynamics, and the fluctuation-dissipation relations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
Note that in fact, thanks to the canonical partition theorem it also holds \(\langle V\partial H/\partial V\rangle = \langle p_V^2/(2m)\rangle =1/\beta \).
- 3.
To see that define the new function \(U^*(\beta ,P)=U(\beta ,\beta P)\). We have:
$$\begin{aligned} \left( \frac{\partial U}{\partial P} \right) _T&\doteq \frac{\partial U^*}{\partial P} = \beta \frac{\partial U}{\partial (\beta P)}, \end{aligned}$$(5.21)$$\begin{aligned} \left( \frac{\partial U}{\partial \beta } \right) _P&\doteq \frac{\partial U^*}{\partial \beta } = \frac{\partial U}{\partial \beta } + P \frac{\partial U}{\partial (\beta P)} = \frac{\partial U}{\partial \beta } + \frac{P}{\beta } \frac{\partial U^*}{\partial P}= \frac{\partial U}{\partial \beta } + PT \left( \frac{\partial U}{\partial P}\right) _T . \end{aligned}$$(5.22)When expressing quantities in terms of T instead of \(\beta \), one similarly obtains \(\partial /\partial \beta = -T^2 \partial /\partial T\).
References
Jarzynski, C.: C. R. Phys. 8, 495 (2007). https://doi.org/10.1016/j.crhy.2007.04.010
Campisi, M., Hänggi, P., Talkner, P.: Rev. Mod. Phys. 83(3), 771 (2011). https://doi.org/10.1103/RevModPhys.83.771
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Campisi, M. (2021). The TP Ensemble. In: Lectures on the Mechanical Foundations of Thermodynamics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-87163-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-87163-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-87162-8
Online ISBN: 978-3-030-87163-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)