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1 Reports Analysis of Temperature fluctuations in ideal gases: From the, Medellin, Colombia ORCID : doi: /RG Abstract Temperature is a physical concept of significant importance in our daily lives. Throughout history, mankind has made big efforts in understanding temperature as an objective property of matter. These efforts led to the interpretation of temperature as a measure of the average kinetic energy of the molecules in a body. Although temperature has been traditionally regarded as a macroscopic property of matter, it will be shown that analog concepts can be introduced at microscopic (local) and molecular scales. The implication of introducing temperature concepts at small scales is that fluctuation becomes relevant and therefore, temperature can no longer be considered as a deterministic variable. The analysis of temperature fluctuations is performed on an ideal gas, following a Maxwell-Boltzmann distribution of molecular speeds at K and kpa. Keywords Fluctuation; Ideal gases; Maxwell-Boltzmann distribution; Multiscale; Temperature 1. Macroscopic Temperature Our bodies contain a huge amount of neural receptors, located on the skin and inside many different organs, which allow sensing hotter as well as colder bodies as a mechanism of protection. These specific neural receptors called thermoreceptors [1], provide only a relative, subjective sense of temperature. As a result of this subjectivity, between the 18 th and 19 th centuries, several different scales were proposed for providing an objective measure of the temperature of a body.[2] After the kinetic theory of gases was developed, the concept of absolute temperature finally arose. Only then, it was possible to understand the absolute temperature as a measure of the average kinetic energy of the molecules in a system.[3] After 08/03/2017 Reports (1 / 9)

2 the definition of the scales, temperature has played a key role in the understanding of several physical and chemical phenomena taking place in Nature. So important became temperature, that it allowed a new scientific discipline devoted to the study of the phenomena related to thermal changes, Thermodynamics, to appear in the 19 th century.[4] For Thermodynamics, temperature 1 is considered a macroscopic property of the systems. In a previous work,[5] it was shown that temperature is a measure of the average thermal kinetic of a system, which is determined from the relative speed of the molecules as follows: where represents the temperature of the system, is its average thermal kinetic energy, is Boltzmann s constant (it is basically a conversion factor), is the expected value operator, is the mass of molecule, is the speed of molecule relative to the macroscopic motion of the system 2, and is the total number of molecules in the system. In macroscopic systems, is a very large number, usually larger than. From the mathematical definition of temperature (Eq. 1.1), and considering that the molecular speeds of an ideal gas can be described by the Maxwell-Boltzmann distribution, then the temperature of a pure ideal gas can be expressed as a function of the standard Maxwell- Boltzmann distribution as follows:[5] (1.1) (1.2) where is the mass of a molecule of gas, and is the average relative molecular speed of the gas, calculated as: (1.3) Given that the expected value of,[5] then Eq. (1.2) becomes: (1.4) 1 From now on, the term temperature will refer specifically to absolute temperature. 2 The relative molecular speed corresponds to the molecular speed when the system is at rest. 08/03/2017 Reports (2 / 9)

3 From which the following equivalence results: Thus, the average relative speed of the molecules in an ideal gas can be expressed as a function of the molecular mass and the macroscopic temperature of the system. (1.5) 2. Microscopic (local) temperature So far, temperature has been considered as a macroscopic property of systems with a large number of molecular components. However, the definition of temperature presented in Eq.(1.1) does not provide any mathematical constraint on the number of molecules present in the system. Such constraint results only from the limitation of our measurement instruments. Therefore, it is possible to define a more general concept, the local or microscopic temperature as: where is the number of molecules in any arbitrary local region (or neighborhood) of the system. Given that the there are many different possible combinations of molecules from the system found in the neighborhood, should be regarded as a random variable. For ideal gases, expressing Eq. (2.1) as a function of the standard Maxwell-Boltzmann distribution results in: (2.1) Now, considering that the local region belongs to a system at macroscopic temperature then, from Eq. (1.5): (2.2) The expected value of the local temperature is then: (2.3) 08/03/2017 Reports (3 / 9)

4 This is a consistent result as the whole system was considered to be at temperature. However, given that the local temperature is a random variable, it will fluctuate around its expected value, and this fluctuation can be characterized by its variance: (2.4) The previous result is obtained because.[5] Eq. (2.5) indicates that the variance in local temperature increases proportionally to the square temperature and inversely proportional to the size of the region considered, that is, to the number of molecules in the neighborhood. As the size of the region increases, the variance in the local temperature decreases. If the size reaches the macroscopic scale ( ), the variance is practically zero. That is, the local temperature behaves as a deterministic variable. A relative measure of the fluctuation is the coefficient of variation, defined as the ratio between the standard deviation (square root of the variance) and the expected value.[6] For the local temperature variable, the coefficient of variation is: (2.5) (2.6) Here, it can be seen that the relative fluctuation in local temperature is only a function of the size of the region considered. Even though these results correspond to ideal gases, the qualitative conclusions obtained remain valid for any other system. 3. Molecular temperature It was previously shown that the local temperature behaves exactly as the macroscopic temperature when the region considered is large enough. Let us now consider the opposite situation when the region considered is the smallest, that is, when. Given that we are considering only a single molecule, let us define this as the molecular temperature : 08/03/2017 Reports (4 / 9)

5 where and are the mass and relative speed of the molecule considered. (3.1) For ideal gases, Eq. (3.1) can be expressed as: (3.2) with the following properties: (3.3) (3.4) (3.5) These results indicate that: The expected value of the molecular temperature of any molecule in a system corresponds to the macroscopic temperature of the system. The relative fluctuation in the molecular temperature for an ideal gas is constant, with a value of 81.65%, which represents a large relative fluctuation. The molecular temperature fluctuation is the largest fluctuation possible for any local temperature determined in the system. From the definition of the molecular temperature (Eq. 3.1), it is now possible to express the local temperature and the macroscopic temperature as a function of, as follows: (3.6) and 08/03/2017 Reports (5 / 9)

6 Please notice that the definitions presented in Eq. (3.6) and (3.7) are also valid for mixtures of molecules and not only for pure compounds, as the molecular masses are no longer involved. (3.7) 4. Temperature fluctuation at different scales The previous sections showed that the expected temperature is the same at any scale considered, from the molecular to the macroscopic scale. On the other hand, it was also shown that the fluctuations in local temperature depend on the particular size of the region. For an ideal gas, the average volume containing a certain number of molecules in a local region of a system at pressure and temperature is. Therefore, from Eq. (2.5): (4.1) Assuming and to be constant, the standard deviation in the local temperature of an ideal gas will be inversely proportional to the square root of the average local volume considered. For example, Table 1 shows the standard deviation of the local temperature in an ideal gas at normal conditions of and, as a function of the size of the region considered. The results presented considered a minimum region size of one molecule. Figure 1 shows the effect of length scale on the relative fluctuation in local temperature (coefficient of variance). It is observed that the relative fluctuation in local temperature is only significant for regions with length scale below 100 nm. Larger regions behave as a macroscopic body. This result has a significant impact on the thermodynamics at the nano-scale, as the temperature can no longer be considered as a deterministic but as a random variable, and the effect of fluctuation should be taken into account. For additional clarity, a Monte Carlo simulation[7] was performed for estimating 1000 different measurements of local regions with different number of molecules. These results are summarized in Figure 2. 08/03/2017 Reports (6 / 9)

7 Relative fluctuation [%] Analysis of Temperature fluctuations in ideal gases: From the Table 1. Fluctuation in local temperature of an ideal gas at for different region sizes. and Total number of molecules Average local volume Reference length * Standard deviation of the local temperature [K] Coefficient of variation [%] m 3 1 m L 46 cm L 23 cm L 10 cm ml 1 cm L 1 mm nl 0.1 mm pl 10 m fl 1 m al 100 nm zl 10 nm yl 3.4 nm * The reference length is defined as the length of the side of a cube of volume 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 1E-09 1E-07 1E-05 1E-03 1E-01 Length Scale [m] Figure 1. Effect of length scale on the relative fluctuation (coefficient of variation) in local temperature of an ideal gas at and. 08/03/2017 Reports (7 / 9)

8 Figure 2. Monte Carlo simulation of the effect of region size on the local temperature of an ideal gas at and. Blue dots represent the 1000 realizations performed at each size. Red dots indicate the mean value of the realizations at each size. The first series from left to right corresponds to the molecular temperature. Conclusion In this report, a brief discussion about the temperature of a system observed at different scales has been presented. The concepts of local temperature and molecular temperature have been introduced to provide some clarity about the meaning of temperature as a measure of the average molecular kinetic energy of a system. By considering the ideal gas case, it was possible to determine the expected value and the fluctuation of the local temperature at different length scales. These results also indicate that even for systems at constant macroscopic temperature, at the nano-scale (<100 nm) temperature fluctuations become significant and should not be neglected. Therefore, a different, non-deterministic approach is expected when considering temperature and thermodynamics at the nano-scale. 08/03/2017 Reports (8 / 9)

9 Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Jaime Aguirre (Universidad Nacional de Colombia). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Schepers, R. J., & Ringkamp, M. (2009). Thermoreceptors and thermosensitive afferents. Neuroscience & Biobehavioral Reviews, 33(3), [2] Romer, R. H. (1982). Temperature scales: Celsius, Fahrenheit, Kelvin, Réamur, and Rømer. The physics teacher, 20(7), [3] Halliday, D., Walker, J., & Resnick, R. (2010). Fundamentals of Physics, Chapter 19. John Wiley & Sons. [4] Müller, I. (2007). A history of thermodynamics: the doctrine of energy and entropy. Springer Science & Business Media. [5] Hernandez, H. (2017). Standard Maxwell-Boltzmann distribution: Definition and properties. Reports doi: /RG [6] Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning. [7] Rubinstein, R. Y., & Kroese, D. P. (2016). Simulation and the Monte Carlo method. John Wiley & Sons. 08/03/2017 Reports (9 / 9)