Introduction

Thermal analysis uses heat as a reaction probe and heat itself is inextricably linked to the development of civilization as a source of warmth and subsequently as a technological origin for energy exchanges, work and industrial processing of materials [1]. The turning point was the discovery of the steam engine and its use, including efforts to improve its efficiency, which led to the creation of a new discipline—thermodynamics, derived from the Greek words thermé (heat) and dynamics (force). Heat, Q, is understood as energy in transfer to or from a (let us say plain—a mass of matter only, in the sense of classical thermodynamics) thermodynamic system, which quantity Q, however, is not a state function of the system, whereas the quantity energy is a state function of the system [2]. So that we can resolve it as a kind of shareable thermal energy, which is just another expression for internal energy revealing itself by the temperature of the system and some thermodynamists would like to banish the word heat as a noun and use it only as a verb (see the word shared), which means to heat something. For that reason, a new variable is introduced as ratio of heat Q divided by temperature T, called the thermodynamic entropy, S, [J·K−1], which is the constitutive measure given be a system’s thermal energy Q per unit temperature T that is unavailable for doing useful work which is able to be produced or generated by the inner imbalance of the system which imbalance causes the ordered molecule motion producing this work, considering the actually possible thermodynamic processes from the point of view of their irreversibility. It is because work gainable is obtained just from the ordered molecular motion, while the amount of entropy is also a measure of the molecular randomness of a system.

The thermodynamic entropy, S, has its macroscopic \(\left( {S \sim \frac{Q}{T}} \right)\) and also statistic worth (S = k·ln \( \widetilde{P}_{{\text{W}}} \)). For a particular macrostate, W, it is defined by the number of microstates or the thermodynamic probability \( \widetilde{P}_{{\text{W}}} \) which is the number of various combinations of particles in individually various energy states within the given W), thus yielding in this given macrostate W with k being the Boltzmann’s constant (1.38 × 10−23[J·K−1]). The k is the proportionality factor that relates to the average (relative) kinetic energy of particles in a mass of matter (plain system, e.g. gas) with the common thermodynamic temperature T of the system [2, 3].

In general, we can use the definition form \(S\mathop = \limits^{{{\text{Def}}}} k \cdot \log_{{\text{z}}} \mathop {P_{{\text{W}}} }\limits^{\sim }\). Then, the quantity S is measured in the thermodynamic unit Boltzmann when K = k, k is Boltzmann constant and z = 10 and for K = k and z = e it is the unit Clausius. Also, it is expressible the information unit bit when K = 1, z = 2 or nat when K = 1, z = e or in Hartley when K = 1, z = 10 is used. Then, it is valid that k·Hartley = Boltzmann and k·nat = Clausius.

In the original material formula S = k·ln \( \widetilde{P}_{{\text{W}}} \), instead of particle distribution \( \widetilde{P}_{{\text{W}}} \) of the given W itself, the message encoding method is used just being the number c of possible ciphering and then the proportionality constant, K, becomes either in information units nat[= ln (e)] or in today´s applied information units bit[= log2 (2)]. It provides a ‘linkage’ between information value via coding and thermodynamic entropy via the system’s complexion so that and we can formally write, when the Schrödinger’s notion of information, I, is taken into account

$$ I_{{{\text{nat}}}} = I_{{\text{o,nat}}} \cdot e^{{\left( { - 1.45 \times 10^{25} } \right) }} \;{\text{or}}\; I_{{{\text{bit}}}} = I_{{\text{o,bit}}} \cdot 2^{{ - \left( {2.1 \times 10^{25} } \right)}} $$
(1)

stated in the information units nats or bits where I[·] is the measure of the order (information amount) contained in the system considered and when Io,[·] is its initial value.

It is worth noting that the increase in information entropy manifests itself as a negative sign in the exponent indicating a loss of information.

At this place, Table 1 of the thermodynamic and information definition of the term entropy is presented;

Table 1 Analogy of thermodynamic and information entropies
Full size table

Also, it is worth to note that the Schrödinger’s and the Shannon’s notions of the term entropy are in the inverse relation, from which Eq. (1) follows.

Thermodynamics in information theory

Let us notice that the term thermodynamic or Clausius entropy holds the units as the heat energy per temperature degree and it stands, however, in a certain contrast with the pure mathematic way using logarithmic relation performed by Shannon. In fact this contrast was between the entropy defined phenomenologically, Clausius, and that defined informationally, Shannon, is erased by the microscopic complexion studied by Boltzmann. It is possible be understood just as the bound [4] realization of the Shannon’s abstraction. The information entropy quantity as a certain measure of structurization or not-structurization (it depends on the approach and point of view) of something, has been related to the term information in the sense of being informed by a message. And, following the well-known paper by Shannon on [5] but without a more detailed definition of what really the term information itself is and what basic properties it bears in the physical or material sense. Our table above is the contribution to that problem. Consequently, we can assume that the change in entropy or change in order consequently may be brought about not only macroscopically by changing the heat content of the system (e.g. the plain one), but also by altering the micro-organization of system’s structure, which, in other words, is actually adding or removing order or gaining the better or the worse information or structure.

In this paragraph, [basic literature is [4,5,6,7,8,9,10,11,12] and further [13,14,15,16,17,18,19] was investigated and applied as a certain unifying description of the basic results of Thermodynamics and Information Theory. Assuming that the heat (Clausius or Boltzmann) entropy is a thermodynamic realization of the (bound) information (Shannon) entropy. Under this supposition, we construct a cyclical, thermodynamic, average-value model of an information (Shannon) transfer chain. We do it just in the form of a general heat engine, in particular of the Carnot engine, reversible or irreversible. A working medium (a thermodynamic system transforming input heat energy) of the cycle is then considered as the thermodynamic, average-value model, as the realization the information transfer channel. It is proved that in the model realized in this way the extended II. Principle of Thermodynamics is valid and we formulate its information form. More generally, we could use an arbitrary kind of directly shared energy. The cycle simulates or models an information transfer process in the channel and enables this way the input messages be transferred repeatedly using transformations their energies.

Our derivation based on the thermodynamic-information consideration about a heat cycle demonstrates that it is impossible, in such a type of channels, for the bound information, contained in an input message to be transferred without its (average) loss, even when the ideal case of a noiseless channel is considered. This loss of information is the necessary condition for any repeatable transfer of messages. This channel is then described by a transformer, medium of input heat, which has non-ideal properties.

We introduce bound [7] information entropies on a system of stochastic quantities realized physically; their values and expectation values are changes of energies reduced by temperature and then give (changes of) thermodynamic entropies. Bound entropies of our physically realized observation, input, output and conditional entropies are, as those free ones, associated in the channel equation. This equation, in its information form, represents an information description of a cyclical transformation of heat energy of an observed, considered informationally, measured system. It is its the most general formulation.

The idea of comparing the basic structures of Information Theory and Thermodynamics, based on the Table 1, is represented by the following Figs. 1 and 2.

Fig. 1
figure 1

Information entropies H (·), H (·|·) and transinformation T (·, ·) on information transfer channel

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Fig. 2
figure 2

Schema of Carnot cycle

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We are in convenience with Brillouin [7], Landauer [9], Gershenfeld [10]; An average information ∆I is recorded, transmitted or computed, ... at the temperature is Θ and then there is a need for a (minimal) average energy ∆W (now ∆W = ∆QW ),

$$ \Delta W \ge k\cdot\;\Theta \;\cdot\Delta I, \quad k\;{\text{is}}\;{\text{Boltzmann}}\;{\text{constant}} $$
(2)

For any information transfer channel, see Fig. 1, the law of entropy (information) conservation, the channel equation or, also, the symmetry of transinformation, is valid,

$$ \begin{aligned} H(X) + H(Y|X)~ = & ~H(Y) + H(X|Y) \\ T(X;Y) = H(X) - H(X|Y)~ = & ~H(Y) - H(Y|X) = T(Y;X) \\ \end{aligned} $$
(3)

The Carnot engine, reversible or irreversible, see Fig. 2, with working temperatures TW ≥ T0 > 0 is considered now as an isolated system with a whole entropy \( S_{\mathcal{C}}\), transforming heat energy ∆QW ∼x∈A to the output mechanical energy \(\Delta A[^{\prime } ] \sim y \in B\) ∼ y ∈ B (input/output messages) as the Shannon transfer chain [X, \( \mathcal{K}\), Y]. The changes of the thermodynamic entropies of system \(\mathcal{L}\) in cycles \(\mathcal{O}\) reversible or \(\mathcal{O}^{\prime}\), irreversible wit the inner friction in medium \(\mathcal{L}\) [expressed in information units (Hartley, nat, bit)], are considered to be the information values of contained in any message on inputs and on inputs of a ‘Carnot’ or (thermodynamically) described transfer channel \( \mathcal{K}\) or, respectively, of the whole Shannon transfer chain [X, \( \mathcal{K}\), Y].

$$ \begin{aligned} \;\;\;\;H(X) = & \frac{{\Delta Q_{{\text{W}}} }}{{kT_{{\text{W}}} }}\;{\text{the}}\;{\text{input,}}\;\;H(Y) = \frac{{\Delta A^{{\left[ \prime \right]}} }}{{kT_{{\text{W}}} }}~\;{\text{the}}\;{\text{output}}\left( {\mathop = \limits^{\Delta } \;\Delta I^{{\left[ \prime \right]}} } \right), \\ H(X|Y) = & \frac{{\Delta Q_{0} }}{{kT_{{\text{W}}} }}\;{\text{the}}\;{\text{loss,}}\;\;H(Y|X) \sim \frac{{\Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }}\;{\text{the}}\;{\text{noise}},~\;\;\Delta Q_{{0{\text{x}}}} \ge 0 \\ \end{aligned} $$
(4)

An irreversible Carnot cycle \( \mathcal{O}^{\prime}\) running in the medium \( \mathcal{L}^{\prime}\) and considered as a thermodynamic, average-value realization or, as such, as the model of an information transfer process running in a channel \( \mathcal{K}\) with noise, the heat ∆Q0x > 0 is drained off from the medium \( \mathcal{L}^{\prime}\). Now we have \(H(Y)\; = \;\frac{{\Delta A^{\prime } }}{{kT_{{\text{W}}} }}\; = \;\Delta I^{\prime } ,\;H(Y|X) \sim \frac{{\Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }} \ne 0\) and then, following the channel equation,

$$ \begin{aligned} T(Y;X) = & \frac{{\Delta Q_{{\text{W}}} - \Delta Q_0 - Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }}~ - H(Y|X)~ = ~~~\frac{{\Delta Q_{{\text{W}}} }}{{kT_{{\text{W}}} }}~ - \frac{{\Delta Q_0 }}{{kT_{{\text{W}}} }}, \\ & \qquad \qquad \qquad\qquad \qquad \; \; \; H(Y|X) = - \frac{{\Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }} < 0, \\ T(Y;X) = & \frac{{\Delta Q_{{\text{W}}} - \Delta Q_0 - \Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }} - \left( { - \frac{{\Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }}} \right)~ = H(X)\cdot\eta _{{{\text{max}}}} = T(X;Y). \\ \end{aligned} $$
(5)

These relations are the same as those stated for a noiseless transfer. We see that

$$ \begin{gathered} \Delta I^{\prime } = T(X;Y) \quad {\text{when}} \quad \Delta Q_{0{\text{x}}} \; = \;0, \quad {\text{then}}\;\Delta I^{\prime } \; = \;\Delta I, \hfill \\ \Delta I^{\prime } \; < \;T(X;Y) \quad {\text{when}} \;\Delta Q_{0{\text{x}}} \; > \;0,\quad {\text{then}}\;\Delta I^{\prime } \; < \;\Delta I. \hfill \\ \end{gathered} $$
(6)

can be considered to be the information formulation of Kelvin’s and Thomson-Planck’s theorem and thus of the Carnot’s theorem. Therefore they represent the information formulation of the II. Principle of Thermodynamics [16,17,18,19]. Our channel is the subtractive one.

Because the value ηmax is the maximum of a set of efficiencies η, it is obvious that \(T(X;Y)\mathop = \limits^{\Delta } T_{{{\text{max}}}} (X;Y)\; = \; \, \Delta I\) valid for the reversible case (∆Q0x = 0). And, because it satisfies the definition of information capacity just as the maximum (supremum) of a set of transinformations, then the transinformation T (X; Y) is the information capacity \(C_{{{\text{T}}_{{\text{W}}} }} ,T_{0}\) for the given extreme temperatures TW and T0. Consequently, on the set of all heat engines with these extreme temperatures, we define

$$ CT_{{\text{W}}} ,T_{0} \;\mathop = \limits^{{{\text{Def}}}} \;T(X;Y)\; = \;\frac{{\Delta Q_{{\text{W}}} }}{{kT_{{\text{W}}} }} \cdot \frac{{T_{{\text{W}}} - T_{0} }}{{T_{{\text{W}}} }} $$
(7)

Further we have the following claims for the change \( \Delta S_{{\text{L}}}\) and for the change \( \Delta S_{{\text{AB}}}\) (in the system (\( \mathcal{{AB}}\))), within the change \( \Delta S_{{\text{C}}}\) of the heat entropy \( S_{\text{C}}\) of the whole irreversible Carnot engine,

$$ \begin{aligned} \Delta S_{\mathcal{L}} = & \oint_{\mathcal{O}{^{\prime \prime}}} {\frac{{\delta Q}}{T}} = \frac{{\Delta Q_{{\text{W}}} }}{{T_{{\text{W}}} }} - \frac{{\Delta Q_{0} + \Delta Q_{{0{\text{x}}}} }}{{T_{0} }} = - \frac{{\Delta Q_{{0{\text{x}}}} }}{{T_{0} }} \\ = & k \cdot H(Y|X) \cdot \beta ^{{ - 1}} < 0,\beta = \frac{{T_{0} }}{{T_{{\text{W}}} }},T_{W} \ge T_{0} \\ \underline{{\Delta S_{{\mathcal{AB}}} }} = & - \frac{{\Delta Q_{0} }}{{T_{W} }} + \frac{{\Delta Q_{0} }}{{T_{0} }} + \frac{{\Delta Q_{{0{\text{x}}}} }}{{T_{0} }} = \underline{{\frac{{\Delta Q_{0} }}{{T_{0} }} \cdot \eta _{{\max }} + \frac{{\Delta Q_{{0{\text{x}}}} }}{{T_{0} }}}} \left[ { = \frac{{\Delta Q_{W} }}{{T_{W} }} \cdot \eta _{{\max }} + \frac{{\Delta Q_{{0{\text{x}}}} }}{{T_{0} }}} \right] \\ = & k\cdot[T(X;Y) - H(Y|X)] \cdot \beta ^{{ - 1}} = k \cdot T(X;Y) - \Delta S_{\mathcal{L}} \\ \end{aligned} $$
(8)

For the resulting change \( \Delta S_{\mathrm{C}} = \Delta S_{\mathrm{L}} + \Delta S_{\mathrm{AB}}\)  of the heat entropy \( S_{\mathrm{C}}\), of the whole irreversible Carnot engine, we have

$$\Delta S_{\text{C}} = - \frac{{Q_{0\text{X}} }}{{T_{0} }} + \left( {\frac{{\Delta Q_{{{\text{W} \mathord{\left/ {\vphantom {\text{W} 0}} \right. \kern-0pt} 0}}} }}{{T_{{{\text{W} \mathord{\left/ {\vphantom {W 0}} \right. \kern-0pt} 0}}} }} \cdot \eta_{\max } + \frac{{\Delta Q_{0\text{X}} }}{{T_{0} }}} \right) = \frac{{\Delta Q_{{{\text{W} \mathord{\left/ {\vphantom {W 0}} \right. \kern-0pt} 0}}} }}{{T_{{{\text{W} \mathord{\left/ {\vphantom {W 0}} \right. \kern-0pt} 0}}} }} \cdot \eta_{\max } = k \cdot H\left( \text{X} \right) \cdot \eta_{\max } $$
(9)

This result is the same as in the case of a noiseless transfer within a reversible Carnot engine. Following from Eqs. (6, 9) we can thus immediately derive

$$ \Delta S_{{{\text{C}}}} - k \cdot \Delta I^{\prime } = \Delta \left( {S_{{{\text{C}}}} - k \cdot I^{\prime } } \right) > 0 $$
(10)

The inequality Eq. (10) reveals that the result change \( \Delta S_{\mathcal{C}}\) of the heat entropy \( S_{\mathcal{C}}\) of the whole irreversible Carnot engine stated in Eq. (9), together with the output average information \(\Delta I^{\prime } \) satisfies Brillouin’s [7] extended formulation of the II. Principle of Thermodynamics,

$$ {\text{d}}\left( {S_{{{\text{C}}}} - k \cdot I^{\prime } } \right) \ge 0\;{\text{or}}, \; {\text{stated}}\;{\text{informationally}},{\text{d}}\left[ {T\left( {X;Y} \right) - H\left( Y \right)} \right] \ge 0 $$
(11)

It becomes obvious that we can state

$$ \Delta I - \Delta I^{\prime } \; = \;H(X) \cdot \left[ {\eta_{\max } - \left( {\eta_{\max } - \frac{{\Delta Q_{{0{\text{x}}}} }}{{Q_{{\text{W}}} }}} \right)} \right]\; = \;\frac{{\Delta Q_{{\text{W}}} }}{{kT_{{\text{W}}} }} \cdot \frac{{\Delta Q_{{0{\text{x}}}} }}{{\Delta Q_{{\text{W}}} }}\; = \;\frac{{\Delta Q_{{0{\text{x}}}} }}{{kT_{{\text{W}}} }} > 0 $$
(12)

Therefore noise output, by its draining off from the transferred information diminishes the output information value, from its maximum

$$ \begin{aligned} \Delta I = & T(X;Y) = H(X)\cdot\eta_{{{\text{max}}}} \;{\text{to}}\;{\text{the}}\;{\text{less}}\;{\text{value}} \\ \Delta I^{\prime } = & H(X) \cdot \eta < T(X;Y), {\text{Carnot's}}\;{\text{theorem}}\;{\text{(the}}\;{\text{second}}\;{\text{part)}} \\ \end{aligned} $$

In our case of carnotized information transfers running in what we can term Carnot’s transfer system (the Carnot engine as a thermodynamic, average-value realization, or model, of a Shannon transfer chain) we have the following

$$ \begin{array}{*{20}l} {\Delta I^{\prime } < T(X;Y), \;{\text{when}} \; \eta < \eta_{{{\text{max}}}} , \;\Delta I^{\prime } \; = \;H(X)\cdot\eta , } \hfill \\ {\Delta I^{\prime } \; = \;\Delta I\; = \;T(X;Y), \;{\text{when}} \;\eta \; = \;\eta_{{{\text{max}}}} , \;\Delta I^{\prime } \; = \;H(X)\cdot\eta_{{{\text{max}}}} } \hfill \\ \end{array} $$
(13)

In both reversible and irreversible cases, we can state

$$ T\left( {X;Y} \right) = H\left( X \right) \cdot \eta_{\max } ,{\text{Carnot}}{\text{'}}{\text{s}}\;{\text{theorem}}\left( {{\text{the}}\;{\text{first}}\;{\text{part}}} \right) $$
(14)

is valid. Because both the I. and the II. Principle of Thermodynamics holds, it is obvious that the thorough transfer of any input message x ∼ ∆QW with the (average) information value H(X), expressed by

$$ T(X;Y) = H(X) $$
(15)

is only the limit (η → ηmax → 1), but not achievable in reality. As in the reversible case, our heat transfer process completes its run with the addition of \( \Delta S_{\mathcal{C}}\) to the whole thermodynamic entropy \( S_{\text{C}}\), and for the average information ∆I′ gained from H(X), we have

$$\Delta S_{\text{C}} \ge k \cdot \Delta I = k \cdot H(X) \cdot \eta_{\max} \ge k \cdot H(X) \cdot \eta = k \cdot \Delta I^{\prime} \ge 0 $$
(16)

The equality \( \Delta I^{\prime}\) = ∆I is valid only in a reversible transfer system in which no heat dissipation, generated from non-ideal properties, so where ∆Q0x = 0 exists.

Our thermodynamic and information derivation, based on a model heat cycle, demonstrates that in the type of the channels considered it is impossible, for the bound information contained in an input message be transferred without its (average) loss. Such information transfer can be worsened by the (heat) dissipation of energy, which means by noise heat (∆Q0x > 0) generated by the irreversible processes in the channel [described by a transformer \( S_{\mathcal{L}}\) of input heat, which has non-ideal properties (inner friction, inertia)]. Simultaneously the whole thermodynamic entropy of the extended isolated system in which this process is running increases, in comparison with the noiseless one, and the maximum of value of the output transferred information diminishes more.

Information and thermodynamics comparability

In addition to the gradual recognition of the effects of heat and its analytical and technological use, it is clear that the history of mankind is simultaneously affected by a certain transfer of knowledge just as an allocation of information (the term “in-forma-tion” is created on the basis of the Greek idea of interconnection of the four basic elements of earth, air, water and fire/heat via forma). Therefore, it would not be possible to build a coherent theory dealing with the reality of cognition (e.g. thermodynamics) without informational influence.

In the context of thermodynamics, what we call information to be given precisely by the difference between entropy/disorder or the internal indistinguishability that the observed system had in its internal structure before receiving the thermally modified energy considered above, carrying information determining the new internal organization of the system and thus forming its better organized state. The previous state in the sense of entropy/uncertainty/disorder or as said its internal indiscernibility is changed by receiving this constitutive energy and the system becomes more structured/organized. The inserted energy is contained or stored precisely in the new structure and the related structural change is measured by the amount of information.

All of this can be described in terms of information as a measure of the structure of the (internal) system or the negentropy of the structural growth of the system, which expresses and says how much information/structure/order there is in via given structurally constitutive event as a result of acceptance constitutive energy. For information values tied to the quantities of thermodynamics, the apparatus of information theory is now used, which deals with the transmission and reception of messages (energy states carrying information, note that syntactic). Note that the increase in the information of this receiver reduces the uncertainty of the receiver about the source of the messages by the received information ∆I, from the initial maximum \({\text{value}}\;H_{{{\text{Receiv}}}}^{{{\text{Max}}}}\) to the \({\text{value}}\;H_{{{\text{Receiv}}}}^{{^{{{\text{Max}}}} }} - \Delta I_{1}\) value in the first round and further to \(H_{{{\text{Receiv}}}}^{{{\text{Max}}}} - \Delta I_{1} - \Delta I_{2}\) etc. Until the resulting HReceiv = 0. Note that we are now talking about building the structure of a system by supplying energy to that system. We will show that only the negentropy of energy across the system considered accompanying this energy supply has the effect of preserving part of this input energy during the growth of the organization or structure of the system. Only in this sense is the input energy constitutive.

As for the plain physical/thermodynamic system the process of its better structuring (e.g. model system system vapour-ice) we must also ensure the negentropy flow across the system, but now just by its cooling, which, on the other hand, requires the energy costs and accompanying heat dissipation from the cooling driver and is resulting in the (thermodynamic but not only) entropy/disorder growing in the cooled system’s environment. It is just at the case when the information/entropy quantity (see the same mathematical definition, via known Boltzmann or Shannon) value is greater or equal to the absolute value of this negentropy flow. Heat pumped off the cooled system is bearing out the original system’s disorder/(thermodynamic) entropy and represents the part of needed input energy for the cooling driver.

Consequently, the information needed to organize for example a mole of water from the chaotic state of steam into a state of perfect crystal of well-structured ice, which would require, for instance, an input of about 35 bits per a single molecule. One may even calculate that an entropy change of approximately 6 J K−1 is required to bring about the required loss, on average, of one bit per molecule, so that we can agree to fashion a general relation:

$$ J/K = 10^{23} $$
(17)

The negentropy flow across a system ensures the stability of the cooled system’s structure and as such is preventing the system from its dissipation and the final equilibrium with its surroundings/environment—from the disorder and death. Also the (thermal/thermodynamic but not only) distinguishability between the cooled system and its environment is growing up. However, at the costs of the (thermal/thermodynamic but not only) disorder in the system’s surroundings/environment.

Let us emphasize that the direct delivering energy to the system but without ensuring the negentropy flow across this system, is resulting in its entropy/disorder growth and death by its destruction—its inner structures are not able to conserve this energy (melting, explosion, etc.) We may disorganize any system or structure by directly applying heat or alternatively by otherwise disordering its structure upon hypothetical ‘withdrawing’ information of its ordering or, in other words, by broken its inner structure (e.g., the very sense of the all military campaigns planning, [7]).

It is interesting to follow Wikipedia saying that the more certain or deterministic an event is, the less information it will contain so that information is an increase in uncertainty we must add this clarifying note: the more certain or deterministic the structure of the system is, the less information is gainable from this system used as the source of messages (which are just the various inner states of the system measurable from or transferable to the environment of the system, to the receiver). Such messages will contain less information just in the sense of the less astonishment or shock for their receiver and this information received causes the less decrease in information/uncertainty or in insufficiency of knowledge of the source on the receiver’s side. On the other hand such a system itself is better or well organized, not giving great astonishment or shock about its inner structure or its outer behaviour (messages sent/measured). It is also when we know exactly the system’s organization, our uncertainity of the system is zero—the system is not able to astonish or shock us. So, in these both cases, the system is not usable as the source of various, more or less expected, messages.

It is important to note that the arrangement is a desirable property of the stability and functional reliability of the system as its particular construction or the aspect of the attainable goal of, for example, our thermoanalytical study. Our resulting knowledge of a studied/measured or even observed system (including studied literature) or a certain well-constructed device then represents well-build systems that save constitutive energy, whether of a machine or a building. This means that these structures not only represent a resource but also save energy costs including the effort required to achieve them, i.e. the desired state of knowledge/functionality or stability.

So we can redefine the term information entropy as a measure of the system’s so called disorganization, its undistinguishability—the uniformity of the system’s inner structure, or also of the system itself and its surroundings. Just as the our own uncertainity about a system’s inner structure or its relationship towards its environment (we do not or cannot know what the system will say about itself—e.g., we are at the very beginning of our studies of a certain but for us unknown branch, literature or book, anything is now awaitable for us) just as the amount of the our missing information (of what the book is, e.g.) needed for us to be informed (learned). Said physically-informationally, to determine the microstates/states/messages for a given macrostate/state/source of messages (for we could know what the studied branch, literature or book is saying, e.g. how the machine is constructed) or to determine strictly the system itself and its environment. For instance, our uncertainity arises when we want determine or distinguish the individual particle in the equilibrial system by its individuality which does not exist, e.g. in the sense of their average velocity.

So, can we afford a deeper way of approaching the central question "why does the Second Law of Thermodynamics apply" in the sense of "why does the missing information increase with time when the system is developing spontaneously in time"? Therm spontaneously means the diminishing of the energy potential differences within the system itself, the very reason for the Second Law of Thermodynamics validity.

Rather than puzzling over why heat flows the way it does, we now wonder how nature defines the questions that can be asked and the number of answers that can be given to those questions, i.e., the extent to which the answers are some way known, and how these things change over time. So when we analyze a situation that may challenge, but seemingly, the Second Law of Thermodynamics, we have a way of looking at them in terms of whether there is some aspect of the system that becomes better defined without compensating our information/structural grow by the loss of information/structure somewhere else. Although this is not possible such an approach opens new avenues for better understanding these systems possible and possible better their behaviour description and, also, for the visions expressed sharper just through this Second Law of Thermodynamics applications.

It was Maxwell [20] who laid the groundwork for connecting information with thermodynamics by hypothesizing a mechanism for violating the Second Law Thermodynamics by a mystery “demon” who could control and separate differently moving molecules. Brillouin [4] further showed that for that “demon” to acquire the information ln 2 necessary for the binary decision of stop-go operation it would have to expend an amount of energy no less than kT· ln 2, thereby increasing the whole system’s thermodynamic entropy by at least k ln 2 value.

Information needed to operate the control requires the compensating the expenditures of the demon’s information gain or also its so-called negentropy gain which discovery resulted to the conceptual penetration of information consideration to the thermodynamics to show the inviolability of the Second Law of Thermodynamics, now from speculative microscopic intruders, and connecting somehow the microscopic to the macroscopic system’s properties.

Let us continue to analyse the statistical meaning of entropy, S = k· log \( \widetilde{P}_{{\text{W}}} \), starting with traditional Boltzmann’s investigations of a macrostate W, assumed it as a ‘complexion’ as the number of possible microstates of a system in the macrostate W. It was modified by Schrödinger who suggested in his book [21] that any living organism is “fed upon a negative entropy” flow across it, arguing by that if \( \widetilde{P}_{{\text{W}}} \) of the macrostate W is a measure of the system’s or of the W disorder, its reciprocal,\( \frac{1}{{\widetilde{P}_{{\text{W}}} }} \) can be relatively considered as a measure of order, Ord.

According to Stonier [22] the terms organization or structure are factually a reflections of the order and vice versa. So, organization, measurable by a value of the quantity Ord, and with it associated information, I, can thus be seen as naturally inter-linked. In the first approximation, we can assume a linear, d, dependence so that we have \( I = d \cdot {\text{Ord}}\;{\text{or}}\;\widetilde{P}_{\text{W}} = \frac{1}{{{\text{Ord}}}} = \frac{d}{I} \).

By rearrangement of traditional entropy by \(S = k\cdot{\text{log}}\frac{d}{I} \) and information by \( I = d\cdot{\text{exp}}^{ - \frac{S}{k}} \) we can define a somewhat more fundamental relationship picture between the information contained in the structure of the system, I, and its entropy just as a measure of its inner disorder S. The greater negentropy flow across the system is (S < 0), the more organized the system becomes. On the other hand, the more thermodynamic entropy S is delivered to the system (e.g. by its heating only S > 0) the more disorganized it will be. When the flow S = 0, the so far achieved system’s structure is conserved.

The following Fig. 3 shows these relations among negentropy and entropy flow and the structure/information or disorganization contained in the system dependently on these flows, by Schrödinger [21] and adapted to Stonier [22]. This picture also shows the extrapolated limits of the maximum order (Omega point), and on the other hand, the highest disorder (thermal death) [23]. The negative entropy flow maintains the stabile structure of a sample and also its grow.

Fig. 3
figure 3

Entropy and negentropy flow and in the system contained structure/information

Full size image

Very simple example of gaining and maintaining a structure is the cooling of what is hot or warm—water, melted iron … We have the input temperature of the sample itself and the output temperature of the cooling environment. While the heat gained before is now pumped off away.

from the sample at the lower temperature, the difference between the input and output entropy (S − S0) is negative and the form or structure measured by information I, in the Schrödinger’s sense, higher by ∆I > 0 against that I0 for S0, arises.

Similar should be, very basically, for the Earth’s two daytimes and for its structures, including it to the elementary conditions for life. Furthermore, we as living, must be in a not equilibrium state, our individual negative entropy flow across us is inevitable constantly [21].

The quantity I, the information, is the measure of the inner structure or order of the system (given by a density of particles and a velocity of their movement within the system’s volume, by their distribution and the system’s temperature), while the quantity ∆S measures the entropy flow carried by heat flow across the system, two temperatures, that of the system and that of its environment, are considered. When this entropy flow ∆S is at the zero value, S = S0, there is no difference between the input (the system’s own) and the output (the system’s environment’s) entropy, now S0, the system is not changing its structure, is just in equilibrium with its environment, the value S0 defines the value I0. When this entropy flow changes by a change of S against a certain and given S0, the structure of the system changes too. For S < S0 the negative difference ∆S = S − S0 represents the negative change of the entropy flow, relatively to the status with S0, and the grow of the sample’s structure, measured by ∆I > 0 arises. As in the refrigerator, the system or sample is bettering its inner structure by the difference of its inner, and as such the input, heat with its higher temperature and with its own entropy, and given just by this heat is being pumped off away, now at the lower output (for the system or sample the environmental) temperature of the refrigerator. And, for the positive difference ∆S = S − S0 the opposite situation arises, we will have a stove.

Let us notice now that though written text can be different due to various alphabets/languages (with various syntactical/Shannon information), the read, the educating, information itself, it means as read and comprehended by the reader (in the semantical sense) remains the same.

In a macroscopic or in the classical thermodynamics point of view, following the Clausius definition of the thermodynamic entropy (thermodynamic quantity for a plain thermodynamic system—seen as mass of matter only) versus the microscopic Boltzmann/Shannon point of view also allows for a curious and rather mismatching description of their mutual relation or "reciprocity".

So let us imagine combusting a book (not suggesting to do really) with a mass of about 0.5 kg (of course together with the mass of black colour of all letters) and with specific heat, Cp = 107 J kg−1. It provides burn heat of about the Q = 5.106 J value. From the point of view of the requirement, we can also be interested in the text of the content of the biblical—semantical information or message, different from the syntactical information of the rows of letters. The second one is our case and could develop in the form of the informational "heat" in the sense of thermodynamically interpreted Shannon entropy of the text.

We can assume that the text holds about a million combinations (n) encoded by using 64 (◦) letters, which can be used to the energy calculation with the Stirling’s formula

$$ k \cdot \mathop P\limits^{\sim } = k \cdot \frac{n!}{{\left[ {\left( {{n \mathord{\left/ {\vphantom {n \circ }} \right. \kern-0pt} \circ }} \right)!} \right]^{ \circ } }} \cong k \cdot n \cdot \log \circ = k \cdot 10^{ - 10} \;{\text{kJ}} $$
(18)

The result shows that there is an a certain incomparability of these two values obtained in this way, i.e. ∆QBook∼106 J >  >  > ∆QText∼10−13 J, the difference of which approaches up to 19 orders of magnitude. This lies far below any detectable fluctuations and would become responsible for simultaneously burning an unimaginable amount of 1012 books.

Nevertheless, we can understand that the ∆QBook represents really the effort and energy needed for writing texts of this amount of books.

Another possible question is where is the heat representing the energy needed for writing the text itself and burned just as the amount of black colour included in the mass of the book and in the text form. It means the question where the energy k· ln \( \widetilde{P}_{{\text{W}}} \) per one microstate in the form of part of the text really is. The answer is that this energy was consumpted and dissipated proportionally at the far away time of writing this text. (The term dissipation now includes also the disorder grow in the environment of this far writer as the natural phenomenon of building any structure, now of the text itself.) None of the biblical meanings or semantics or education sense of the text is, in the Shannon/Boltzmann way, considered.

Measurability of heat using thermal analysis

The statistical distribution determining the entropy also carries with it the kinetic energy distributed over the inner movement of the individual participating particles. Its final value is the resulting state of heat transfer and dissipation to and from the system providing a balanced state called equilibrium, which can be statistically described by a weighted quantity termed temperature. Its meaning and value serve as a basic thermodynamically intensive quantity that aids the definition of all basic thermodynamic relationships. Therefore, the definition of temperature declares that it is a measure of how hot and cold a body is, and it’s simply the average rate of kinetic energy, Ek, for the particles in a given substance. It means that the greater their kinetic energy, the higher the temperature of the body. Specifically, \(T = \frac{{2E_{{\text{k}}} }}{3kT} \), where k is the Boltzmann constant (1.38 × 10−23JK−1). Of course, heat input and output are needed to change the temperature changing thus the kinetic energy of the movement of the particles involved. This is evidently subject to the Newton’s law of inertia of the mechanical momentum of each particle and thus the temperature change due to heat flow cannot be instantaneous but has a time delay due to the mechanical inertia involved. Of course, there must be an adequate number of particles in the studied substance for the enough true averaging statistical validity. If there is mutual competition between particles in the volume and on its bounding surface, then it depends on the ratio of the number of particles to each other [24,25,26] and leads to a significant influence on the transformation which temperatures are in relation to the reduction of the grain size.

Heat changes, which lead to a significant influence on the transformation temperatures in relation to the reduction of the grain size, does not bring any information, it only increases or decreases the amount of disordered movement. When this random particle motion is organized into usable mechanical work by directionally ordered movement, a special machine is needed, which can be called an information transducer, and which does not change during such a process, but only wears out. In the case of non-informational heat, this machine is reactively complex (e.g. a steam engine), if the thermal information is transformed into an improved form, e.g. electrical energy, then the processing machine is simplified (alternator).

Heat can also serve as its own reagent in a discipline called thermal analysis, which deserves a few notes of practical information. Heat thus serves as an information source of practical evidence about its own effect on the state of the object (sample) under study. Heat that does not carry information brings an increase in the kinetic energy of the particles, which at a certain temperature culminates in the structural reorganization of the particles into a new arrangement. It becomes energetically different and thus associated with the release or absorption of reaction heat, which manifests itself as a phase transformation that can be indicated by the device. The consequence is also the information gain from knowledge about the attainments of transformation, i.e. its location (temperature), size (calorimetry) and change dynamics (reaction kinetics). The sample under thermal study de facto serves as a disposable information transducer, which is structurally altered during measurement. Thus, the practical conversion of heat into an experimentally measurable quantity can be achieved by means of instrumental monitoring of the temperature behavior of the studied sample at its controlled surrounding temperature (thermostat, furnace). The research adaptation to an experimentally measurable quantity can be achieved by and instrumental monitoring of temperature behavior, e.g. using an informer—thermometer sensor. Then it turns out to be one of the most important methods for detecting and thus describing thermal changes and is based on the fact that heat itself acts as its own executive. Thus derived method of thermal analysis (TA) is, unlike X-ray diffraction, a destructive technique that studies the relationship between material properties and its temperature when the sample is heated or cooled in a controlled manner, when heat acts as a reagent. The central motive of such a measurement is the recording of thermal effects caused by heat transfer and its dissipation providing widely detectable and all kinds of transformation sensitive data. This process was well described by Newton [27] and shows that it is not instantaneous but involves both a time delay and the inertia of the material to temperature changes. The literature deals with two expressions of thermal inertia, i.e. its dynamic and static connotation. In dynamic form, it occurs due to heat dissipation during all thermophysical measurements, which arises due to changes in the heat capacity of the sample owing the changes in the mechanical motions of its inherent atoms/molecules. It has two macroscopic forms, integral (known as the calorimetric constant [28]) and differential (whose specificity shows the non-linearity of the background s-shape of the DTA peaks) [28, 29] derived via the classical Newton time derivative of temperature, as

$$ q = K\cdot(T_{{\text{h}}} - T_{{\text{c}}} ) = -\, C_{{\text{h}}} \left( {\frac{{{\text{d}}T_{{\text{h}}} }}{{{\text{d}}t}}} \right) $$
(19)

where q is the considered heat flow from the hot body (with temperature Th) toward the cold surroundings (with temperature Tc) using a coefficient of proportionality K (thermal conductance) and difference between the rate of heat content changes of the hot body with heat capacity Ch and the cold surroundings with heat capacity Cc when t means time and Ch value is neglected as diminutive. The derivative in the last term of the Eq. (19) has the meaning of the impact of thermal inertia, however inseparable. On the other hand static thermal inertia is generally understood as the degree of slowness with which the temperature of the body approaches the ambient temperature. It depends on the square root of the product: heat capacity C, material density, ρ, and thermal conductivity λ, i.e. up to \((\lambda \rho C_{{\text{p}}} )^{\frac{1}{2}} \left[ {{\text{Jm}}^{{ - 2}} {\text{K}}^{{ - 1}} {\text{s}}^{{ - 1}} } \right]\), which effectively represents the ability of a material to conduct and accumulate heat and is used in describing the thermal comfort of buildings [30].

The resulting state of heat transfer in the monitored sample is a state determined by a quantity called temperature. In this thermally dynamic regime, however, the definition of temperature, which is thermodynamically defined only in equilibrium [2], is violated. In the regime of steady changes (constant first derivative), the classical theorems of static thermodynamics can be applied appropriately enough merely used [31,32,33]. However, for methods using rapid temperature changes, the question residues when and if the classical concept of temperature will remain valid and when a new definition proposed as tempericity [34] of simultaneous determination of temperature and its change arises. It is called the principle of indeterminacy, which is similar to the well-known relationship that the speed of movement of an object, dx/dt, and its position, x, cannot be measured with the same precision of temperature changes, dT/dt, even the temperature of the object, T, itself.

Dynamic thermal conditions require a new adaptation and the introduction of some specialized thermodynamics [33,34,35], where a fundamental step is to maintain the validity of those thermodynamic relationships under the conditions of constancy of the first derivatives, i.e., for example, uniform heating rate, which is also the idealized state of the thermal analysis methodology. However, there are still unresolved problems associated with the practical use and description of methods with rapid temperature changes [36, 37].

This apparently requires a modified methodology due to the practical needs of non-standard technological procedures, which are manifested in new areas of so-called kinetic phase diagrams. These non-included inconsistencies are manifested especially in works on non-isothermal kinetics, when published studies specifically deal with classical modeling reactions at the reaction interface [38] using geometric analogies, neglecting the practical aspects of measurement such as the supply and removal of heat and reaction products from the reaction interface [29, 32].

In particular, the simplified description of the reaction temperature in the sample attributed to the inert reference that follows the introduced temperature program is misleading and requires a new approach even though known from before [31, 32]. All this is represented by more than half a century of habitual use of simplified mathematical models [38] which reappear even in the latest publications and even books [39, 40].

An unsolved question is the informational value of such published papers, where the central motive is often the determination of the value of the so-called activation energy, which is influenced by the evaluation methods used and which in many cases loses its meaning for reactions in the solid phase [41,42,43]. This is related to the general problem of the applicability of the Arrhenius exponential expression in kinetic mathematics [44, 45].

Added to this are the unwanted political aspects of the procedure of publishing and disseminating information, especially the need for the highest possible journal impact factoring, submission politics of top journals, dissidence for some fields and newly introduced opinions, etc., which is a special subject of scientometrics [46].

Some aspects of information policy seem downright laughable, but unfortunately also rather reprehensible when pushing scientific reality precisely in the field of thermoanalytical kinetics. For example, their some authors forget about scientific ethics and becomes a strict supporter of only their personal competitiveness, completely neglecting citations of the concurrent works of their professional rival such as [47,48,49] which should not be reflected in serious science.

It is the area of non-isothermal kinetics that shows the danger of the comfortable practice of using established simplifying procedures [38] that enable the convenient publication of information that may not correspond to reality but which has a good citation response. Their improvement is a challenge to future generations [50,51,52,53], recalling that the increasing volume of publishable data requires some innovative ways of controlling it, which is difficult to implement in the mess of reliable versus unreliable information often contained in publications as discussed in [54].

Discussion and conclusions

Recall that the proposal that the phenomenon or variable "information" should be associated with the concept of "entropy" in the sense of statistical thermodynamics was once put forward as a kind of mathematical joke between Neumann and Shannon. This can be demonstrated by the work of Wicken [55], who called Shannon’s terminological choice of borrowing the word entropy from statistical thermodynamics as follows "… the loose language [of] the dark god of obfuscation". We hope that our presented contribution is a good step on the way to understanding the thermoanalytical relations between the concepts of entropy and heat as independently observable quantities. Another work in this vein is close to our approach and is elaborated in [53]. The practical and theoretical aspects of the actual dynamics of thermal measurement are well specified in [50, 52] showing that the understanding of thermal analysis should be re-evaluated and understood as the analysis of heat/temperature settings by determining heat fluxes relative to the environment, not just the standard examination of the reaction properties of the sample itself. The authors believe that the future development of applications of thermal analysis methods lies in understanding the meaning of the word thermal, which must be interpreted not only in terms of heat/entropy but also via the dynamics of heat transfer [50,51,52]. Related to this is the information worth that is introduced, transformed and subsequently resulting from the completed measurements, which is a novelty not yet captured but relevant in the future of sophistication and automatization.

The above incorporation of the information field is in accordance with the tendency that information will actually flood more and more the contemporary world, including science. It is not incomprehensible, but remarkable that even the scientific world has become the target of misinformation. This can be seen in the situation where the content of a number of professional publications may not coincide with the established socio-scientific approach (mainstream) and when a number of such works deviate from conventional practices and thus head into the so-called dissident area [50, 55,56,57,58] otherwise defined only in politics. Opposing viewpoints, which is a dissenting approach, is always essential to the advancement of science because disagreement is the foundation of advanced peer review and is important for uncovering unfounded assumptions and problematic reasoning. Allowing and encouraging disagreement also helps generate alternative hypotheses, models, and explanations. Even in the area of thermal research discussed above, we can encounter entities that are not often welcomed by ordinary users—the influence of thermal inertia [29, 51, 59] or tempericity [33, 34] as an unbalanced temperature is typical and our inventive approach involving informational thermodynamics [60] may sound similar in this context.